Transcript for:
Triangle Properties and Theorems

Hello friends. TYT AYT geometry the second part of our subject description book We are on the subject. We will make an angle in the triangle. We continue with the basics. Basis topics are of course almost triangles we can say all of them but especially the first all the subjects as they are if not because we use it in topics Our no-no was true at first. It was hungry You know. Later and now We will make an angle in the triangle. subheadings by separating, e, by dividing into pieces I explained this issue in detail. So sometimes maybe you'll see in the future some topics on the pages of your book even more than the number of pages. Because the first You kind of see it times. So I I count it that way. At least more I don't think I've seen it before. And A basic theme is the triangle. And this is the basis angle in a triangle. Naturally, the right we need to give. And the shapes we need to recognize. In geometry, the most we need to be very careful one of the topics is the subject While learning, yes, of course we will not memorize. but on the other hand, the shapes to recognise. What do shapes offer us? Why are we doing the awesome trio at work? Why do we make isosceles triangles? Steep What does the triangle do for us? What is Pythagoras? does it work for us? So most of the time as I said before These are the questions we will ask, what are we doing? will be useful and how can I get this information? It means I can use it. Therefore Now let's introduce the angle in a triangle. Now ABC is a non-linear 3 point AB, BC and AC are straight lines triangle ABC is the combination of its parts We will say, friends. Now of course Let me tell you something about this. Well, Because I looked at it before I remember. Mustafa Yağcı's triangles There is a video titled eee. There asks students about such unique definitions. Here it says come on A BC triangle Let me give you a definition. What is a triangle? TO What do students say? Here is the truth triangle is the union of three non-existent points It is called. Mustafa Yci was doing something there Am I remembering incorrectly? Look, there are three I connected the dots. Here, is this a triangle? Not. So, yes, there We have three non-collinear points, but them as a line segment by combining the combination of these three We're forming a triangle, friends. This the name of your shape is of course undisputed it will be a triangle. We have a triangular polygon. What is the first property of a triangle? Inside the triangle the sum of the angles is 180°. Our other feature The sum of the exterior angles of a triangle is 360° Friends. And one in a triangle The sum of one inner and one outer angles in the corner is 180. So A + D is 180. The sum of one inside and one outside 180 B + E. The sum of one inner and one outer is 180 C + F. Therefore, these three pieces of information questions very nicely We will use it, friends. Immediately some I'll just prove it later in places For the curious but it's easy around here It will not tire you because there are proofs. Come on, let's look at the interior angles of a triangle. Why is the sum 180°? With one method Let me prove it to you. Because some rules with more than one method we can prove it. Look at the bottom like this uh, so the base is like this on the ground, you know? Let's say it's a plane, like this If we consider the base as a ground plane If we say here is a parallel to the base I pulled. What good will it do me? Look If I say alpha, it is alpha because of Z. Theta If I say theta because of Z. Come on over there too Let's call it beta. Be careful. Alpha + beta + What does theta mean due to the concept of a straight angle? was equal? 180° at right angles. TO On the other hand, the interior angles of the triangle are also alpha, beta, theta, right? Then what happened? Here are the interior angles of a triangle We proved that the sum is 180°. This much. I've said it before It's not like in mathematics. Some Our proofs are really simple in geometry. Now we said that. TO These are already CF, AD because of the correct angle and BE 180° no further proof is needed we won't hear. The sum of the exterior angles Now let's talk about the sum of the exterior angles Why 360°? the sum of the exterior angles of a triangle? What Do you know what you can do? 180 - A D instead Summer. Write 180 - C instead of B. Instead of E B E 180 - Write B. Look, I'm collecting the exterior angles. only. Now 540 - A + B + C in parentheses what came? 540 - 180 to seriously 360° It happened. The sum of the exterior angles of a triangle. By the way, this is what we call convex shapes like quadrilateral and pentagon external shapes such as hexagon, heptagon, octagon all concave shapes have exterior The sum of the angles is 360 degrees. degree. Let me give you this information. Convex what is so concave? As if inward like he got punched like that. Here is an example is a concave quadrilateral. Or a Let me draw a polygon. Ok. Look at this It's a slightly concave polygon like this. It is curled inwards. Therefore, this Let me also say that. But we in geometry only concave within the curriculum We have one quadrilateral. All the rest We are dealing with convex things. Well When I say polygon, it means like that not with things that are curled up correctly. With such classical polygons as you know We will deal with it. Yes. What have we learned now? The sum of the interior angles of a triangle is 180, The sum of the angles is 360. And also an inner angle the sum of the exterior is 180. Only this feature Let's look at our questions. Now the EU parallel DE BAD 50° DEC 70°. Accordingly I'm asking you about the ECD angle. Where is ED? Yes. This angle I showed in red how can it be done? You immediately say, teacher I'm here because of Z. Look, because of Z. Sir, of course I will stick to 50 degrees. Then the sum of the interior angles of the triangle is 50 + 70 + x = 180, you say, and x is 60° You will find it. In every question, the second way You don't have to bother with it but sometimes I'll give you a little extra geometry like this We are just learning, good comments sometimes in extra discourses to add I will be found. For example, this question is the second way Let's solve it like this. C Let me draw a parallel from the point to the base. Both to the EU and thus to ED I pulled. It became 50° due to Z. Later Pay attention immediately because of Z, it became 70°. Now the image looks like this: 50° 70° Since the entire length of a straight angle is 180 so look at the vertex angle of the triangle 60° It remained. Let this be the second way. Let's come here. These friends are talking about Let me remind you of it from time to time. Please do not skip ahead. I do this I know, I know that Don't go. Because some places yes You may know. I have an objection to him no. But I, my friends, pay close attention to this. I will. Because I created a fiction, I arranged the questions accordingly. things we will always use later Even if it was a simple question, I placed it. For example, why is this question so common? What kind of a show did I do? We are right work on the angles, stop right now the question. Get to work now. Now on the right I continue with the word angles. What were we saying about the angles in the right? Opposite The angles are of course equal to each other. Alpha. Not alpha? Two lines intersect time. That's when I started asking this question time too yes there are two triangles here. This triangle and this triangle. But what? I tell you? A and 70 - A are equal to each other. Naturally A what came from here? 35°. I'm asking about B, but let's find these. Look, let's see. Let's get to know the question. Then please be careful. There is a triangle. A the other triangle with its angle. Even like this Let me show you. If I draw these two triangles again, I drew it like this, like this. Once already because of the opposite angles this is A and this is A. We even took it out already. I found A to be 35 for this place also came 35. E B friends This is B. E, after all, pay attention to this. 35+ B is me to the right angle C -10 If I said that, what should I say about the angle on the left? C -10. Because its two angles are the same. B 35 B 35 E the sum of the interior angles of a triangle is 180 I'm making it up and saying B's Imagine being 100. I mean, let me exaggerate. not be like the answer in the question. Well this is 135 does. Makes 135. Seriously look at the other angles remain at 45 degrees. That's why two If two angles of a triangle are equal, then The third angle will also be equal. Think about it Look, the EU is the other one. E if this is C it has to be the other one is C. Why did we use this? The question is where to put C-10 will you paste it? Here it is you will paste it. Then C-10 is also here did he come? Yes, it came. What do we say now? what about after this? Angle in the line With the help of our red angle C - 10 + 2C + 10 = 180°. From here we get 3C = 180. What happened to C? It was 60°. Now we find C to be 60. If he had asked, we would have answered that too. From this then let's place it. I'm pointing to A. 35°. How many did we find for C? E 60. Subtract 10 and get 50. The sum of the interior angles of a triangle is 180° According to 50 + 35, 85 out of 180 interest. What's left? 95°. Therefore We found the answer to our question. In this way Look, I won't give detailed information. It was necessary. It's simple but from here on out We are responsible. So please Don't be fooled by this simplicity of geometry, Is it possible? Now let's look at this question. What said the question? 40, with D1 being parallel to D2 He asks what is 60x. Get to it now. Yes. How can we do this question? Now of course it was true. At angles too in the sense of let's remember. I'm extending it like this parallel. If the parallel is extended to 60° is 60° from the corresponding angle. Look at the given Where do I link the information? So in a triangle we have the angle. Interior angles of a triangle total 40 60 more 100. Then back what remains? Of course, to get 180 we need x Look, there are 80 left. In this way It was enough for us to see the parallel. Yes. Now let's see. X - Y 50°, What is BAC 40°, Z? Completely basic I'm talking about what we've learned. The triangle The sum of exterior angles is 360°. Then what about him? Let's use it. Because since there is x and y, look there involved involved involved x + y + 140 due to the sum of the exterior angles of the triangle 360° and this way I can now reach the other side If I throw it in, I see that x + y is 220°. And when I add it side by side, 2x = 270° x what happened from here? 135° happened. When X is 135°, what is left for Z? 45° It remained. Yeah, let's see now this to our question. ABC is a triangle. Pointed angles same, the lined angles are the same. Accordingly, deb I asked what the angle was. This is black we'll find the angle that I painted. Now to blue I painted. Or what can we do now? BC I already gave you the angle of 66°. Ok. After that, of course, sometimes in geometry a letter if we don't have to but sometimes If we have to, two letters we can use it like this. Because the unknown there is here. I don't know. A for dots I said, the lines are B. So what? will be? How do I find the angle DB? this given situation will provide? Normally to find this angle actually really, you know, red in that triangle I wish I knew B and A in the triangle I colored Let's do it from the inside angle. time. A + Thousand 2 times 2A + 2B + 66° their sum in the larger triangle is 180. 2A + 2B What came from 180 - 66 to e 114 a + b? 57 came. Now I want to find friends Let me call the angle C. Here too 180 - C Let me say. 180 - C + A + B triangle interior 180° 180's went from the sum of the angles. A + B is 57 so C what came from here? 57 came. Now look at this As you can see, we found C. This is a You can also think of it as a question type. At the same time, sometimes they give us such angles gives seemingly irrelevant. We are these other 3rd party by taking advantage of the total we can find the angle. Now here is another one I will tie it to the ground. You just noticed Is it? Those 180s are gone. Then friends rule is coming. Our new rule. Note this get it. Two inside, one outside. What does that mean? One the sum of two interior angles in a triangle neighbor who is not their neighbor Look at the bottom, alpha's neighbor is theta is the angle. The neighbor of the bet is this Y angle. Naturally these two are called beta and alpha our third non-adjacent angle is the third exterior This is our angle. Therefore Friends, I say two inside and one outside. this in short. I'm not saying this for a long time. So the sum of two interior angles 3rd external that is not adjacent to them for two until you say it is equal to the angle the sum is equal to the exterior angle of the 3rd angle We can say that. So now this question is We found a + b to be 57. Come here now Let's solve it this way. The most important thing for us It is one of the features of two inside and outside. In most questions, two inside and one outside will be given to you. I will show you nice comments. Look at this In the triangle, we actually called this a and that a. Where do 2 inside and 1 outside equal? A + to BC. E C What happened automatically? 57. This feature Please put a star. It will save us a lot. It will enable us to do little unknown and in the isosceles triangle we really work It will help. I'll show you in a minute. Yes. Let's see. Now this new feature we will use. How many times is 84 - a + 16 x? degree? I said. Now find x for yes we need to know clear information a. So how is it, teacher? Look, that's exactly what I said. event. In geometry, angles in triangles or Notice which topic is in the angle questions one inside one outside always without keep it in the back of your mind. For example, this In the question, the sum for two is here equal to the exterior angle. What happened then? 84 - a The sum of a + 16 is equal to 5a. This way the a's are gone. 100 = 5a to a happened? It was 20°. Since A is 20°, 20 + It became 36 from 16. Make sure it is the right angle don't forget. One inside and one outside from 180 - 36 then what came x? 144. Wondering about this There may be those who do. Teacher, why two? the sum of the interior angles equals the third exterior angle equal? Let's see now. This is how I do it. Take it such a simple proof for you. Like this I'll draw a parallel to the base. A lot when you solve everything with parallelism, it's like As if we were going to solve the question with parallelism don't think. We are doing proof right now. From Z was it B because? Yes. A from the convergent perspective did it happen? Yes. Here are two for you Prove that is equal to an exterior angle. This Write down the proof immediately you can. That was it. But A rule that is as simple as it is important. Now try this question immediately. Yours I want you to do it. Let's see how you will think. Let me add a nice comment Me too. The interior angles of a triangle are 2 5 6 If proportional, what are the exterior angles? with which numbers is proportional? Now pay attention to this I will ask. If 2 5 6 is proportional to what is this? means? An inner opening 2k, 1 internal angle 5k and 1 internal angle 6k I can say. Well, the letter is not a problem there. It doesn't matter what's in a and b. But at least the interior angles are like this I showed. Well now here are these If I add it up, it makes 13K = 180. K from here it becomes 180/13. Look, it's exaggerated. That's right, but I don't think it's a good way to go. of course. Open an interior 360/13, open an interior 900/13. Open an interior 1080/13. And what? what will you do? To find exterior angles eee the sum of one inside and one outside is 180. You will go and take them out one by one. First interior angle, exterior angle, sorry. 2nd exterior angle and 3rd exterior angle. And these too I will rate. Something strange happened here isn't it? Then of course this way is right although it is not beautiful at all. What I will do it? I'm asking you about the exterior and interior angles. I gave the angles. Interior with exterior angles What is the relationship between angles? Let me write 2k, 5k Let me write. The total for 2 of course here is equal to an exterior angle. 7K 2. After What was our other angle? 6K then look 2K 6K where else is it equal? External angle 8K. Then 5K What does 6k more equal? To 11K. Seriously 11K. HE time came an external angle 8K. An exterior My opening was 7K. An external angle was 11K. This What are the numbers proportional to? With 8 7 11. So what happened to these angles after all? We know. Constant numbers, that is, direct We don't know but we can find out. But I they already run away from you not just proportional to how much I want you to see that it is. Therefore Look at this question this way and give it a star Let's put it, although it is not difficult. Because two How easy it is with the help of internal and external We got rid of fractions. Look now Did you notice this here too? question by the way? Let me point that out too. What did we find? I'm deleting these. Because it is unnecessary. Look inside the triangle the sum of the angles is 180. The number 13K is 180 equal. 13K 180. See what this is? 13K. Look what this place is. 13K. Look Let's see what this place is? 13K. Is it a coincidence? not? Because there is already an inside and an outside its sum, that is, only one angle What was the sum of the exterior angles next to it? 180. So since they are all 180, 180 Since there are 13K in all of them It is not a coincidence that it is 13K, it is normal. Yes. Now the premise question. ÖSYM sometimes You know he asks. USA and BCF one each If it is a triangle, how many of the following one is correct? Take action now Let's see what you say. The first one is A + B + D + E. You tried isn't it? What if I show it on you? It would be good. A + B + D + E. Oh my God. How can you say 180? Like this Of course, you know, two inside and one outside lover I don't want to act like it but two an exterior is so savior that it is sure My eyes are one for friends, two inside and one outside He's calling me. Provides little known. One you get the job done in total. New twin you find edges or existing You use the property of isosceles. HE That's why it's very important. Therefore 2 inner one outside I came here. D + E 2 hi inner outer What will happen on the sidelines? It doesn't matter. TO + These are in the pocket now. Then reverse From this angle again this is D. Then this I saw it, look. One angle of the triangle is A, the other is angle B. Let me show it in color. Other interior angle D + E. Look what it really is. Question that gives the sum of the interior angles of a triangle already to me. Premise 1: Clear information is correct. happened. C - F C - F is equal to A - E. What relevance now? C - F A - E. So first At first glance it doesn't seem to mean much. But let's try to see. Between them What kind of connection is there? With your permission I'm deleting. You know, we already saw it. Now Let's focus here. Here A, C and E A connection between F. Then this I can say. Look, be careful like this. This Sometimes there are minuses, I'll mention that too, knowingly. is given like this. Actually, a little bit of you pluses because it's easier so that you can use it. Something like this let's do it. Let me move the F to the other side Let me see. Although it is still negative I can't get rid of it. Well then, what kind of I will improve the comment? Look now C and A Look at the connection between. Minus I was going to throw it to the other side. From minus I was going to say get rid of it. I couldn't escape. Each There remains a minus of sorts. In that case I gave up. Be careful. C - A two inside one externally C's difference from A is equal to B Friends. Do you see? Look here Let me show you. A + B = C. Good. In that case C - A became B. Ok. Such a there is a connection between them. Then let's come A - E This relation between C and A I saw. Now let's come to the difference between F and E to the relation. Please be careful. Here again Where is E + B equal to in that red triangle? to F. So B what happened again? It became F-E. HE time friends here F - E and C - A were equal to each other. Available right here If I fix it somehow, take C left to the side. F - C. Take E to the right. What happened from here? TO It became E-A. Multiply F - C minus C - F.E - Multiply A by A - E and negative both sides I multiplied with. That's how I found it. Or not at all Until you do this, there are two types of operations I won't let you do it. Direct A left Let me throw it to the side. A - E. Direct left F Let me move it to the right. C - V. This is easier It happened, didn't it? Multiply by minus again Until I say something like that, here are two drinks for you we benefit from the blessings of the outside an image. Then the second premise is also It was true. Or should I ask this question anyway? the only reason is that there are two inside and one outside to show that it is important. F - E to B equal. E is actually the second premise When we do this, it's a bonus. reward 3rd premise we have already seen. 2nd in the premise F - E. B and E are indeed Since the sum is F, what is F - E really? became equal? B has three premises true. 4. In the premise, as in every question, you know There were or are guaranteed mistakes. This It's something like that because look at this. A + B + C 2C + B why would you take this? A BC E C + C + B. The C + B's are already gone. A what did it equal? to C. So AC is equal to Is it? No. Of course. Pay attention What did we just say here? Look two internal to external is equal to C - A B. HE time C - A 0°? So no, of course not. If B is an angle there and B is 0, there is a triangle Impossible. So clear information is here It is not equal to AC, friends. This It was an important question in some way. Well You know, in mathematics, there are such difficult questions or sometimes important good questions we see. We put stars Friends here in geometry All of the examples are some kind of star importance in terms of what we will use later in terms of comments. For example, in this question definitely very important for us in the long run precious. Where will we use this? do you know? For example, in similarity, For example, in the formula of sine, the structure we create, we create fiction. HE That's why it's so valuable. Let's see a question. We will see in the question what the importance is. X yes, yes, yes. Well, EDC angle, EDC angle Let's place the question mark. Now this How many degrees is the angle? I'll take a look. What's up? Here? What kind of connection are x and y? can I install? There are pointed angles you will say. Where is your place? I'm thinking now. You know, knowingly What I think is a student If we look at it from the perspective of For example, what would I say? If I were to see it for the first time geometry I spent those times as a Let me try to think and imagine. The teacher explained. I say it openly He also likes a bit of geometry if necessary. I liked it at first. I tried so hard. I thought there was no lie. What about this? Teacher, this much internal and external conversation did. Yes, the interior angles of a triangle are important. but it's not that important. Because the interior angles of the triangle, that is, the exterior angles of the triangle The sum of the angles means a lot It doesn't. Because it's already about there What should I ask you directly? Simple very I ask simple things. I already asked But two inside and one outside are very valuable. one by one It allows to write in terms of . In that case Look now sometimes I give you two like this between the sides of the triangle here at an angle as I showed with phosphorescence gives the interior angle of the triangle and is as follows: linearity also works for us like this. There are 2 internal and external parts, so I shouldn't say y. Let me say A. 2 internal and external requirements x + a where is it equal to? to the place I painted red. Then the angle is already pointed A What is left according to? Here I painted it green X left to the angle. As you can see, X has arrived. Beautiful. So since there is an X here now this is an interior angle of triangle ECD At the earth's surface, an angle of X°. Y + X + question sign 180. y + X is already 100°. So that means What is the part with the question mark? It was 80°ce. Look, the question came naturally. Sometimes the first You may experience this feeling at first. You'd think I found it by chance. Problem not. You know, it's like you find it by chance. Yes. But these always have meaning. At work Friends, this is what a stereotype said Let me be yours now. Give two triangles I will show them in color like this with green. One side of these if it gives linearly, especially at an angle If it gives you the angle here, it will give you the message if this is theta angle, two inside and one outside Accordingly, it is consistent in this respect. This Leave a note if you want, friends. We will use this much. We got it done. Let's come here now. We asked, "What is C?" B - A 26. A I gave it. I gave it a C. I gave it a B. 70° There are pointed angles and there are line angles. We'll go to C. Yes. You know this? Of course you can think. Take a look Let's see how you say it. Now I need to find C. It's related to A and B. HE I don't know about the EU at the time. There are 70 there. What relevance? Now this situation two inner and one outer triangles within each other It might work. Always a definite benefit I'm not saying that but there is a high probability that it will work. high. a little bit with a lot of unknown numbers also reduce the number of unknowns provides. For example, if we say a here now Friends, I said angle bisector or a. Two inside if I subtract A from B due to an external condition where does it equal? Pointed angle. Look I don't skip C right away. C because the end The move will find C so easily anyway If I were you, I wouldn't ask about C. Anything else I ask. Think of it this way. So what did I do? I care about others. From there on the road I will go up and reach C. Because I wish this Even though I know the angle, this angle is 70 and C is the interior angles If I could handle the whole situation. In that case What did I say immediately? In this triangle, red If I subtract A from B in the triangle I colored I find the other angle. B - A seriously B - A The sum of A and B is equal to . From the angle bisector It was B - A again. Now from here The question is what gave me B - A anyway? 26. So he says he is not worried. You drink two You've made it outside now. Award 26 degrees I'm giving it to you. It became 26, 26, 52. 52 70 122 more arrived. Interior angles of a triangle Here is the 58° C option from the total happened. Look, questions A are easy, how? etcetera. Ease, convenience, Don't look at the difficulty aside. Let it go you saw it at the right angles. Look, it's hard There are also questions. Same topic, same feature. different types. This is also here two inside and one outside Look how valuable it is. And also as many unknown numbers as possible To reduce it, look at this question too as you can see, they are of different types It makes writing very nice. This right away Let's take a look at it. He asks how many degrees is x. 76x64 There are pointed angles. Yes. Now again one Let's see how we can think. A check Let's. Well, comment. What do you think? I will first guide you on the path I desire, then let me show you an alternative way. Look guys, I need to find X. Do you know how I can approach? to the question? X, you know, I say this first I wish I knew this. Look here It would be nice if I knew. Or else What could I possibly know? So this question If I know the marked part, the interior of the triangle I am handling it very well from their perspective. Well At first glance, it's just my job to find it It looks like it will work. But dotted Why are the angles given? Look, I won't find it either. I thought about what was necessary and this What is the relevance of what is given? Now I'm looking. Pointed angle. Pointed angle. As I just told you, actually It looks like this. Look at that triangle If you make two inside and one outside, let's call it alpha. Where is alpha + 64 equal to? Here. Look I combined the information. I wish I said If only I had found the angle just now. Let me call it T here. If I knew the angle T in triangle AB. Thus, from the interior angles of the triangle, I would. Oh then I said what was given to me Let me check. Let me make 2 inside and outside I said. Because we have a common angle here There is. Be careful because of alpha + 64 drink 2 angle t due to an external alpha + 64. Either There is already alpha here. What's left? 64. So what was it then? 76 + 64 180 things 180 where does it come from? 140 Subtract 140 from 180, leaving 40. Here you go, ask the question first I think. What good is what I am given? I said and we handled the situation nicely. This the first way and really what are the two inside and outside that it is such a useful rule example. Come on, what else can we do? The second way is that you say here again a Let me say. I said alpha, sorry alpha again Let me say. 2 inside and one outside here 64 - Let me say alpha. The alpha of the already dotted angle I know it is. Come on now. 76 I'm writing alpha. Follow. 76 eee 64 - alpha + x is an interior angle. One interior angle and the other inner opening alpha. E= 180. Alphas are gone. What x came from here again? 40. What you see As a second way, in this question, Well, I was able to develop such a comment. This It was also nice. Now we see it, we see this. I first came to the main house Look, I'm drawing the frame. Two inside, one outside It's important to me. Then look again two drinks We did it starting from an external point of view. Another Well, can anything be improved? Each You don't need to think about that in the question, but to think a little bit about geometry at the beginning It is necessary, friends. I'm analyzing shape. Because there are shapes in many formats You know. And the new generation of this There are and my adaptation to each of them necessary questions and what ÖSYM asks The questions are generally standard in geometry. but it breaks the standard in certain places I have a very good command of the subject. I need not to be anxious. Geometry Don't accept anxiety, friends. You will be directly affected. Seeing geometry It's not a job but if your stress increases, yes the seeing part of the job you put it in a situation where it enters. In that case and yes, seeing ceases to be geometry You put it in the food. Now here a I say, I say alpha. Like before. I call this alpha. Let's take a look. You know, sometimes, sir, there is alpha, there is 64. Right there, a theta to the 3rd angle Let's say. I made two lines there like this. Due to the sum of the interior angles of a triangle alpha + theta + 64 sir 180. Alpha + theta Look what came, teacher? Alpha + theta 116 came. Then if alpha + theta is 116, this is If it is 116 then 76 + x = 116 from here again What happened? It's 40. Here are the questions in full 3 We solved it this way, friends. All kinds of but in all three methods there are two inner and outer we used. Sometimes such lettering It also works, as you can see. Let's come here. How many degrees is x? I asked. It is a very classic question type. Let me tell you yes to this type of question. Well, Keep trying, it will save us another time. These are pointed angles, line angles given. These mean they are equal to each other that. I'll tell you from now on. Geometry This is what I do in class In the sense of parallel, the points are equal. If I put a line here, it is equal. You know these? I don't write long, this style is simple in the examples only on the figure we see. What do we say? Dotted dotted I need a striped X. Let me write an x here too. Come on then Let the pointed angle be Y. This time there are no angles. Oh please don't do this. Is it ok? ÖSYM is such an exam moment came and did something. For example, he went you can easily go to the most stylish ones asked a question. Okay, I'm not saying anything but It seems like we can't go out of style Let's act like it. And it's definitely like this If he asked, this is 90, 45, 60, sir, it is empty. Don't say give. Let's prove it beautifully Let's give an example and conclude. Now here we have x and y. Then friends I will make a move like this. The only one This is the void. Shall we call this Z too? Let's say yes. What happened now? X + Y + Z Because 180° is a straight angle. Huh. Later Look, come here. Z Y attention If you do 2 inside outside Z + Y X didn't come in that blue triangle? Yes. Then 2x What is x from 180? 90°. Here is our topic This is it. The event in a relaxed manner We got it done. Is it ok? Yes. The house is beautiful again a question. Four different colored sticks They will have a common corner at the point are combined in the form. Black and yellow colored bars line It's happening. Blue and yellow bars red and black at the angle between angle between colored bars unchangeably shaped like sticks when placed blue with black and yellow bars and green sticks are connected to each other It becomes parallel. Accordingly, the sum of a + b is how many? Now first of all, my topic is What? Of course, there is no doubt about it here. 180° 2a + b became 120°. Triangle e linear ignore 180 degrees because it is We don't. Since it's 60, what's left? remains? 120 remains. Is it true? To the right When you look at it, we have such a comfort There is. If black is parallel to yellow, it is convergent Of course, it won't be from this angle? Yes. Then pay attention to the blue here green parallel to each other. Yellow already continues. Corresponding angle. A with angle B angle corresponding angle. Here it is again thick Let's show it this way. A little more Of course, it shouldn't be that thick. Attention et look parallel here. Then he continues on his way. This at pretty good angles as we know was the equality of corresponding angles. Therefore Of course, what actually is it? It is equal to the EU. Then if I write here instead 3a = 120 As you can see, the sum of a is 40 and b is 40. This is 80°. Now we have one feature There is. Again, two internal and one external born our feature. This is a concave quadrilateral name friends. Curled inwards a quadrilateral. The property of this quadrilateral is this. D angle outside, 3 inside inside equal to the sum of the angles. So here are 3 one inside and one outside are equal. So how is it? Sir, if you say this, it is from an extremely simple place. is coming. I shot this linearly I extended it like this. 2 inner one outer B + C where equal? Equals here. Then B + C 2 inner one I painted the exterior red Where in the triangle is it equal to? B + C + A to D equal. Oh look what came to you, here's the feature. So again two inside and one outside Thanks to this feature, we have also saved We became. But of course I tell you this Let me tell you. I proved it here, but From now on, don't do that. Well you will paste the direct rule and pass These are question types. Ok Is it? There is no need to give it that much importance. It might be a good question, but the rule is Let's use it. Now AB parallel FG, BD parallel angle bisector E BD and DE. BCE 60 BDE I'm asking how many degrees is 20° FE EFG? I can use this now. What? stop it if you want. Look, do you see this? This is it here, my friends, teacher alpha asks. What do these have to do with anything? Please do this be careful. If only we knew these dots here Well, look at this and that already When we know where we actually contribute provides this? Contribute to finding the Alpha provides. Because I do the zigzag rule. So what should I say? aa A aa. HE What happened to time? 20 + 2a here is new rule I learned. I just saw it. 20 What does + 2a equal? To 60. From here to what came? It's 20 degrees, right? Yes 20 came. Naturally, look at the green now the angle I painted is 40 because of 2a. Then The angle I colored green is again due to 2A 40. What shall we say next? Outward Ministers will look at 60 and in the same direction. Look, be careful. Here for now x Let me say. Then 40. and 40. Then 60 + x = 40 + 40. From here, what is x from 80 - 60? came? 20. Because its complement is alpha What happened to alpha? It became 160°. Here you go, look at this We handled the incident easily. Do this immediately You try, let's see. Two identical quadrilaterals have one side each black when drawn overlapping the edges are parallel to each other. Accordingly, the angle expressed as x is How many degrees does it measure? Now The black edges are parallel to each other. Where is it? Sir, where is the quadrilateral? Look at the regular quadrilateral since it is not, what do we mean by that? time? Here is the 1st concave quadrilateral and the second concave quadrilateral. What you see Since I said it is identical as well, from here too You can already understand. Where is it, teacher? didn't say. Wait a minute, how? will I understand? Look at someone who is identical someone is red and someone is green one is black color and one is blue color. He Actually, come to the other one now. Green color then red color then really blue color then seriously black color. What you see Identical shapes such as appeared. Identical What are the characteristics of shapes? Congruent shapes The feature is the same for everything. More I don't need to explain the similarities. Think normally. Two shapes if everything is the same as each other are the same. Let me still emphasize this. If you want, you know exactly what the question is about If I didn't see it again according to this emphasis think. Everything about identical shapes is identical. The angles are the same, the sides are the same. This I also said it. I also gave the Tüo. Come now Let's see. Once all the identical shapes If everything is equal, then y is green, sorry. If the angle between black and blue is theta the angle between black and blue is theta wouldn't it be? Then come here. The angle between the black and blue edges angle theta. Here too black and blue angle e the angle between the sides is theta. What happened? Blacks are parallel to each other According to this theta the sum of the theta is 2 theta 180 due to tetma I gave it a 90. Let's put a star on this question. Actually, I wrote this as I am learning. I am developing question. I'm learning a little heavy for its level. I'm improving is also normal for the level. Theta e= 90° arrived Is it? He came. Beautiful. What else did I say? Look at these because they are 3 identical shapes. if the angle between green and blue is beta in the other, the angle between green and blue beta. Why is that, sir? These identical shapes. Then immediately If we make 2 inner and 2 outer in this blue triangle, What does beta equal? 2 inside and one outside To 40. What came in beta? 20. Good. Now I saw that the beta was 20. In its place I'm writing. Tetn is also 90 I have seen. Now come on. There be careful. There's a concave quadrilateral right there. Do you see? It's here too. Even this I can use it. What happened then? This Of course, it's not related to what I just gave you. after all but even this part you can use. 40 + 40 + x 40 + 40 + x equal to what? 90 + 20 to 110 right there equal. What does x come from here? 110 - 30 came out of 80. What am I anyway? I wanted? We wanted this 30 right here. We found X. If we consider it according to school subjects, approximately If we spend it to the fullest, where is the first from second to the last second approximately 40 See the introduction to angles in a triangle in minutes we did it. Sum of two interior and one exterior interior angles completely basic, such as the sum of the exterior angles We have solved our questions with features. Just one extra probe and two insides we find from the outside but concave with the model we used as a quadrilateral We looked. Now we will move on to a new section. That, my friends, is an isosceles triangle. Here It is very valuable to us. Now isosceles We continue our journey from where we left off with the triangle Let's continue. This is a place to rest You can also think of it as. Well, me too I'll be taking a little water break. This with two skener triangles after the break We will continue. Those who listen to one piece he will continue on his way anyway. Well, other Our friends are now giving them some water, We grab some coffee, tea, etc. and continue on our way. we can. See you later. In an isosceles triangle I say. What for those who listen to one piece is happening? We continue, friends, Let's start with the isosceles triangle. As the name suggests, the twin triangle is a two-sided triangle. a triangle consisting of equal sides. We have a top angle here. Of course I'll tell you this in a moment. Well, equal union of equal sides the angle formed at the intersection is called the vertex angle we say. Look at the two equal sides The angle formed at the intersection is the vertex angle. Top the base to the side opposite the angle we say. And let me state this right away. Where is it? The angle at the top, as in Turkish, is the angle at the top. not in the sense of. Look now this is a Let be an isosceles triangle. Equal sides The angle formed at the intersection is yes, this is alpha is the apex angle. Is it ok? And the alpha opposite is the base. After this one When an isosceles triangle is given, the base is not the one at that place, but the normal top angle detect the opposite edge. Twins the edges are already isosceles. These are like this Let me tell you. base angles are equal we will say that it is. First of all, this It's enough to just say it. An isosceles triangle. The top angle is different from these. If we say alpha, theta, theta, the base is like this angles are the same. Connecting equal edges becomes our top angle. Therefore, immediately Let's look at our questions. What to pay attention to will we? Once upon a time, friends, this is ours. very valuable for you. In two symmetrical triangles Since the base angles are equal, we get will make a good contribution. On the other hand, new We can find side equations. Suddenly if we intersect more triangles. And this too isosceles is also a topic in the subject we will process but this inside the angle You will also notice that in isosceles. If there are two intertwined triangles, there are two inner triangles We will go outside by doing it. Because little we want to deal with the unknown. Look Now I asked what is x. Even here we can do it. Look, let's get started. 35° since the two sides are equal double line double line. Right at the base What happened to the angles? It was the same. Isosceles because it is a triangle. Then 35 more two 70. Base angles are the same as the inner and outer angles. because of what? After all, this edge I phosphorescent and this edge is opposite each other the same. Double line double line. Base The angles are the same, 70 degrees each. 70 70 triangle How many degrees does the sum of the interior angles add up to x? remained? 40 degrees left. This is how it works in intersecting isosceles, two inner lines Don't forget to make the exterior. Is it ok? Let's come to this question. From the EU perspective The angle of The total is given as 70°. How many degrees is CAD? he asks. Take a look, he asks immediately. So don't dive in headfirst. One check. E wants CAD from me. CAD. On the other hand, I look at the UAE. BE here. I'm looking at E AE. Something like this I am a dotted angle. I'm looking at AEB. Like this Let me call it a linear angle. What does that have to do with anything these? Now if he didn't give it to them, what are they? that is. The edges are like this right now at least represented by different symbols. Then the intertwined isosceles Making two inside and one outside triangles is not enough for us contributes through the unknown. Comment I will do it. Find my angle in red I actually kind of need this for There is. Look, if I knew these question-marked places or if I knew the totals, look, this is it It is possible. Either know C and D separately or Know the total of the CD. All kinds of me Does it lead to the red angle? Yes. HE I immediately paste the alpha. Isosceles triangle. The base angles are the same. Alpha. I use 2 inner and one outer. Alpha. Alpha. What are their totals? Rye equals 2 alf'ya. I call this theta. I say theta. A different triangle is given. 2 inside and 1 outside Where did 2 theta become equal to? Here. So, friends, what do alpha + theta mean to me? was given? 70. What happens to 2αfa + 2 theta? 2 when you take the multiple of 140. Then the red angle Here is the sum of the interior angles of the triangle. This in the triangle 2αfa 2 theta 140 isa remaining What will the angle be? It will be 40°. Look at this In this way, the concept of two inside and outside again we used. We will continue to use it. Here, as I said, two inside and one outside to use these questions I'm not writing. In these questions there are two inside and one outside I used two inside and one outside I care. Or something like that We are not obsessed, friends. Where is it? I am ultimately a teacher but Our profession is teaching. So in our lives Oh, I love math. Well, there you go I'm like a math lover I'm not behaving. So mathematics is my life is located in a part of it. Teaching It is a part of my life. But I mean, sometimes things to love come to me, you know? Even questions can be loved. Ok, good. We say questions etc. but eee loving It's not something to be done, after all. Yes. Let's see now then x e here There were numbers in the previous question. Here it is There isn't even a number of flights. Special angles now Sir, is it definitely one of 30, 60, 45? Of course, not in this question. There is no such thing. But it's one of the shapes we'll use a lot. What can I do? Just one more tip I will give it to my friends. Two triangles inward when it passes from the little ones like this Saying alpha alpha and making two inside and one outside it might make sense. Look, it makes sense. It certainly makes sense. Always do this not necessarily, but it is highly probable. When we analyze the shapes this tells us What does it contribute? Head over it Let's tire. Look, the questions are already there anyway. It's not difficult but we will use them. When you do this, there are two in this blue triangle. when you make an inner and outer automatically Look at it like this to get 2 alphas You do. Then I'll throw it here If it were theta or isosceles, here too Look, if it is isosceles, this question has something like this We offer 2 alpha contributions. Did you see? Look at me, both inside and outside, at once took it to the letter. Even with a single letter I did. Let's see then I'll delete this. Now you can make a note of it if you want. Let's look at this exact question. Introvert There are isosceles triangles. Then also He gave the following information. Phosphorize it Let me show you. With the EU If you notice, BC is also equal. So big It was a triangle with angle X broken down here angle C angle in total again angle X. TO What will happen now? Immediate questions Instead of focusing on what is given, I look at it. I'm making two inside and outside. Alpha. Alpha Let's say. As I just said pack up and go. 2 inside and one outside here Paste 2 alphas. Ok. Then the base angles are the same. Which triangle is here? letter Let me give it. ACT triangle 2 alpha isa X ne happened? It was 2αfa again. Meanwhile, big What did we say in the triangle? Of course this is A angle C is the same as angle. Then Angle 2 angle C is also 2αfa. Our remaining angle and here is the other angle on the top left. remained alpha. Then now in the grand triangle If you pay attention, all angles are alpha 2αfa Since there are 2αfa, the sum of the interior angles is 5 the total is 180° alpha what came from here is 36° came. What does the pipe want from me? 2 alpha So x, then we found this to be 72. This In the figure, there is a special triangle in 36 72 It is a triangle. It brings the rate to gold. Edges the ratio between the gold and gold ratio It is a triangle that provides. Under I think I know exactly what that is. look it up on the internet. Then we already we will also talk. Now our topic I won't distribute it because it doesn't exist. ABC triangle. AB is equal to AC. BD is equal to BC. How many degrees is B? Now let's see it in colors Let's see. Once we equate AB and AC I gave it. Let me make this a little thicker. Then BD I gave it equal to BC. Look inside How can we use a passing triangle? Now I'm going to be strategic here. What we said? Going from small to big makes sense, right? From small to large What do you do when you leave? Fewer letters allows you to use. Okay here now Be careful, you can do both inside and outside. required. If you start the question here, it is wrong. Is it possible? It certainly wouldn't be wrong. But like this, where do I make two inside and one outside? Or do I have to? Look here This too may be a matter of debate. Do we have to do this, sir? Nope. Two inside Can't the question be solved without an outside? Well I guess it can be resolved. Let's try, So we say let's see. Look something like this I did. Alpha alpha at least I am very I didn't like it. TRUE. Why drink it? Do you know if it is not digested? I actually I'm throwing alpha alpha here to anyone who says beta up to or alpha x15 this is better I saw. Look, I say, it's more beautiful. More I'm not saying it's true. Start small. Two make an inside outside. Go alpha here say alpha. Alpha + 15 too. Look how beautiful it is. Here we put two inner and one outer alpha + 15 paste. E isosceles triangle from here too paste alpha + 15. Beautiful. By the way this in between the large triangle is also isosceles If angle C is alpha + 15 then of course B angle is alpha + 15. Naturally DBC alpha remained at the angle. Then the hill angle alpha, base angles alpha + 15 happened. And so as you can see 3αfa + 30 180° alpha 50° arrived. What is the question from me? wanted? B AC. Look this way I found the question. I prefer it that way. I'm growing like a snowball. Two inside I'm going to do an exterior. Inward like this question recently Look, the angle I can use again. This place I used it and got here. Because here too If I know here, I can also write here with the same letter. I'm naming. Then I'm here focus and find this place two inside and one outside I wanted. This is the way that seems better to me. But sir, it is already an isosceles triangle. If you say "Wouldn't he give it to you?", look, it will happen. This is not problematic in the question. It's very nice. Secondly, I called alpha, head here. Even Let's call it a different alpha, this time theta. The interior angles of a triangle are the same as the base angles since it is all theta What remains? Theta -15. An isosceles Since it is a triangle, the base angle BDC is the same theta. Look, it's very problematic here. So there was no situation. 2 more theta theta 3 theta -15 = 180 theta what came from here? Since 65 theta is 65, the base angles are same 65. What is left at the top angle? 50°. Here is the second answer to your question. Now ABC and EBD are triangles. AE to EF equal to each other. FCCD = US 75°. There are isosceles triangles. Isosceles In a triangle, of course, the first is angle equality Okay, okay, but sometimes there are two to make an exterior, the angles as much as possible writing in terms of each other, less known use in both mathematics and geometry It works for me. So what should I say? 1 let's call it like this. I started from here. Start from the top if you want. Secondly, the foundation our knowledge of right angles. No different using letters. From the opposite angle it is already a. From the inverted isosceles triangle again a and two inner What is the external angle C equal to? To 2A. This concludes our question. 3A + 75 180 from the interior angles of the triangle. A from here happened? From 105/3 35°. What does the question ask of me? X to 180 subtract 35 from 180 to complete. 145° The answer to the question was done. Yes. Look, try this too. DB = AE AD = DE = EC. What is the angle DCB? I asked. Where is 36° now? given? ACD 36 was issued. Yes. This is how it became 36 here. Ok. Later Based on this, where is DCB? in between? DCB angle. I'm asking here. I'm asking about such a place. What can we do? Now this question is actually relatively easy. Do you know why? Extra oh oh oh Oh, it won't be a one-letter situation. Because I gave the angle. Follow this time. 36. Since the base angles are the same, 36.2 interior 72 from an outside. The base angles of two triangles are the same. de and AD. Then it's 72 again. Okay. What next? Shall we do it, sir? AE and DB are very unrelated isn't it? Let me show you with AE like this. Isn't DB so irrelevant? Then again AD where is here with? Well, it's irrelevant, isn't it? Look at it this way. Actually it's irrelevant not. Look here, x edge pattern here y. Here is the x side and here is the y side. X+ Y. X + Y. What are you going to see? Actually this the length of side AB of the triangle AC the length of the sides is the same. For this reason Since the top angle is 72°, 72 out of 180 interest. 108 Divide by 2. B de 54, C de 54 happened. Then if I subtract 36 from 54, I get How many degrees is the angle he asked? It's 18°. That's the thing, friends. This too Let's tick it like this. actually taking advantage of a different situation there you notice that the image is isosceles we should have. Friends in an isosceles triangle We sometimes encounter length questions we have a draft shape coming up. This is equivalent I immediately said that if it comes, we can use it. I placed it as a note. Well Lettering is important to us. Use as few letters as possible is important. In isosceles triangles, the vertex is taking advantage of the same bases You can name it with letters. Example 2xte what will happen? Right here 180 - 2x to 2 If we divide, 90 - x 90 - x shares. Indeed, the sum of the interior angles of a triangle is 180 It is possible. Again, this is not 2x If x then what will happen? Base angles 90 - x/ 2 90 - x/2. The geometry to be aware of the subtleties of friends required. For example, this is a subtle important is an image. Is it ok? When it comes to the same you use it, it will be useful to you. where is geometry Oh sir, there are always new generation questions You know, that geometry, that classic to connect to the new generation through We need these very much. Imagine being a master of something. to be aware of all the details shouldn't you be? Somehow the game There are students playing. So no game not playing computer games or any game or skill Say whatever you need to say something you're interested in in your life has been. At least in the gaming sense. The better you want to be, the better You don't have much knowledge of the details of that job. It's necessary, isn't it? So what and how You know better what to do. Well, I was going to get into food conversation but whatever I won't enter. Then we do something. Now We can't get out of there. At least Let me tell you. Finally, friends There are people around you who cook very well people. People who cook very well not just because he's talented anymore what to use, where and how, how to adjust the gold, that is, will learn to talk to various objects until I actually know the details of that job you see that it is. So matching they can do it. So sometimes a different When something happens, he can say this. what about me I'll pair this with this. Such a beautiful a product emerges. Actually, look, If we apply it to geometry, the same thing happens. different, interesting, new, what kind of question When he comes to me, I am old those questions based on what I learned I will solve it. So this is for us This is an important image, friends. Sometimes It is necessary to make such lettering. Let's look at this question for example. A = 2B shown in red as it is What is the measure of the angle in degrees? Just this the note I showed to explain thing. Look what happens if you write 2b instead of a? 90 - b 90 - b come here right away. And what Is it ok, teacher? What does that have to do with anything? 90 in spouse - B. B what else does he do? 90. Then the top angle is also What happened to the triangle? 90°. Even seeing this sufficient. Because it's a right triangle or something, look. While I was here, "Wait a minute, teacher, what does that mean? You say there is. I put this in a right triangle I'm placing it. On another topic I'm placing it. What are we going to do there? Pythagoras will sometimes accompany the similarity. Just finding that 90 degree angle is a lot. valuable in itself. Let's try this question right now. Same thing on a rectangular board colored edges are of equal length Two triangular plates are hung on it. Blue edge of the plate on the left parallel to the top edge of the board. On the right the red side of the triangle is point O when it rotates 30° clockwise around parallel to the bottom edge of the board The measure of the angle represented by X is I ask how many degrees it is. Yes Let's try and think together. Now friends, the edges are the same color It was equal. Then I'll be a little more If you want to emphasize, again Let me show you. Look, they are the same color. I do something like this. Look at these blue ones then they are equal to each other. Then the reds are equal to each other. He Look here friends, the next topic I won't say anything about it. To this Please accompany this without knowing the similarity you even need to know. There are two triangles. The edges of these exactly the same. Look, do you see? The edges are exactly the same. No edges the angles of two triangles are the same could it be different? I don't know, teacher. If you say it's possible, you're wrong. you are thinking. Because the shape is the same. This shape your lengths to achieve you need to change. Oh sir, where are you from? you know? The angles may be different. HE Do you know what happens when you look? I bought Let's say this is a stick. Like this I went and made the stick like this. does not intersect. It intersects elsewhere. You can't get it. If you claim to have obtained geometry We shake. So think about it There is a triangle. The angles are different. Look 3 4 5. There is another triangle. I'm throwing this away angles 37° 53 here is 90. The other one's angles 30 60 90. This is also 3 4 5. It is not possible. Pieces of that length anyway When combined, a single triangle is formed. There is no other way a triangle is formed. So, that clarity is two truths. the gap between. What is that clarity? possible? With the combination of those lengths possible. Naturally, friends accompany each other there is no need even. Later in the short we will see again. Everything of the same shapes Let's say it is equal. It's simple We will use the information in the question. For that reason I emphasized. What happened then? Isosceles triangle base angles are the same. Blue, blue These are the same triangle. Red on the right the sides of the colored triangle are the same for this would be xx. Look here right now I placed it. Such base angles are the same after all and the same triangles. Then what what should I say? He said the question is you. Let's show it in color again. A it will be difficult to turn it. Like this Let me do it. Let me take the red one more. Look what you did with that red? Not clockwise? Where was it saying? him? Yes. Clockwise. Clockwise You turned it like this. How many degrees? Like this I turned it around. Look. Even dotted and dashed Let me do it. I turned it like this. How many degrees I turned? 30°. What happens when I rotate it by 30°? dead? This is parallel to the base. In that case Since friends are parallel to the base The blue edge was already parallel to the top edge. Rule Z has arrived. Look, be careful. Z rule have you seen? I'm showing it again. Z like this became the rule. Then due to X + 30 Z where does it equal? So that's equal to X + 30. In this way, all three angles of the triangle were found emerge. x X x + 30 when I add 3x + 30 180 X what came from here? It's 50°. This is the situation, friends. Actually As you can see the solution to the problem is simple but what should we pay attention to? One here peer images. The second is parallelism. Why given in parallel? No empty information is given already in new generations. Use that information It is given as follows. And of course, ÖSYM again Friends, see the question in the form the information is fully explained in it. It's not much of an image did not show. So let's put our star here. Yes. Now this place is very important for us precious. Being an isosceles triangle One of the conditions is actually friends. I wouldn't say it's one of the conditions of being a thing. pardon. Being a double triangle one of the results. It is also very important is the result. An isosceles triangle vertex a perpendicular from the point to the base If you lower it, it will directly split the shape in two. So the height of the isosceles triangle is is the axis of symmetry. The axis of symmetry is So let me tell you friends. Information Known as. You take a shape when you close it, it overlaps if it comes and there is no space left If you see only one piece, that's right. is the axis of symmetry. This works. Or Friends, look at something now. I will show you. There is a rectangle, okay Is it? This divides the shape in half. not the axis of symmetry. Why does he know? Are you? What happens when you fold it do you know? This is how it happens. For that reason the axis of symmetry of the line that divides each half not. When you cover it completely each other without leaving any empty space if the covering shape is obtained then the symmetry axis naturally this height symmetry is the axis. He knows what's going on anyway Are you? Pertaining to the base of a triangle height both the angle bisector and the side bisector. I briefly I say burn. Long long sometimes height I may not call it angle bisector or side bisector. Know now. Burn it, friends. height angle bisector side bisector. Isosceles where in the triangle? Only the vertical that goes down to the base divides the base in half and the angle bisector is the same in time. Is it ok? Or the side edges not. So if it's an isosceles triangle, here The one you sent down has no such duty. We just got it down to the base. Because direct divides the shape in half. What you see I wouldn't say this right here either. required. Yes, it is an isosceles triangle If I give it, look, I am giving the information. Now because this place is important I will stop. I think it is an isosceles triangle Let's say I told you. Red colors I I said it in the sense. Is it ok? I have these I gave it. Then you ask the question directly to the base if you download it you definitely this middle ground you must show. Definitely the angle bisector you must show. So if it is an isosceles triangle Once I draw the height, the others also There is. Or look now at an isosceles I gave the triangle. What I show in blue I gave. The question told you this or that is a median. I just said it was a median. HE If I give the isosceles time friends median There is no need to ask questions the moment I draw. You just turn 90 degrees here and here You will place the angle bisector. Because height is both angle bisector and side bisector I said. Even if I give one away, the others There is. When? I think it is isosceles if I tell you. This place especially I emphasize. Look, it is isosceles If I tell you, show these differences. Again I drew. I am an isosceles triangle I told you. It was me who gave it, not you. I drew it. One angle bisector here. The question doesn't even need to be said. Angle bisector is also height and median? Yes. Then the question is If I give the isosceles, clear information is the median and is the height. As you can see, the isosceles triangle when I give the height angle bisector the median is a single line We need to pay attention. When you get to the bottom. So, what happens to the ones below, sir? Look Now sometimes I tell you that it is isosceles I won't tell. Look, I said it here. Have you seen? I always gave the same colored ones. First I gave the red colors in the image. I gave the blue color in the second one. In the third I gave the color blue. So that's it The questions I gave are isosceles says. Okay, come here. There is no triangle, isosceles or anything like that here. There is no such thing. Did you see? What is here? will there be friends? A triangle. Look, a triangle. Any Let me tell you, it looks like a triangle. One in the triangle If the height and angle bisector are the same line, then the net information this triangle is isosceles. If the median and angle bisector of a triangle are If they are the same line, then the triangle is clearly isosceles. In a triangle, the altitude and median are the same If it is true, then the exact information is this isosceles. Let's see. Now look at this in the beginning I do not give that it is isosceles. We will think about it now. Really isosceles Is it? Let's comment. Yes. Do you know why it is isosceles? If I say alpha, there is no 90-alpha left here. Is it? Yes. If I say alpha here, 90 - alpha doesn't it stay? Yes. Base angles are the same What is the triangle? Isosceles. So what? In that case? Height and angle bisector and median. This thing, let me take a look here. Here's the image. Oh, I gave all three of them here. Pardon Excuse me. Here are all three already I gave it. This is the one I already gave three of for both altitude and angle bisector and there is no need to even discuss the edge no. This is already isosceles. When? Because height angle bisector side bisector when were they identical? Isosceles. This I told you. I'm passing on this. Come now here. Look here, it's independent of what it was before. At least everything was the same as before he said it was true. But here it is not. Only the angle bisector and the side bisector are the same TRUE. Sufficient. Only median and If the angle bisector is the same as the line, this is clear information isosceles triangle. Come on teacher, prove it Let's look at me. Let me prove it. Come. Now Friends here know what's there Are you? We actually have companion footage. Once from the angle bisector Therefore, there is an alpha edge. Alpha thing angle There is. The side opposite the alpha is the same. The side opposite the alpha is the same anyway. Also, be careful here, there is no common edge. Is it? Yes. Then what is this? Those who don't know it could be. No problem. For proof I'm telling you. Side angle is the side congruence. Are the edges the same? Yes. The angle between is it the same? Yes. Then side angle side cosine of this because of its accompaniment can also be called a theorem. It doesn't matter. Net due to the cosine law or parity information. What happened then? Other edges The same thing happened. Twin edge has arrived. In that case There is no need to extend. Look, the proof is straight where does it come from? From the rules of companionship. Naturally, they are old friends. Same angle bisector and side bisector are one If it's true, there's no need to ask questions. You set the height. Be careful. Because the angle bisector and the side bisector are the same If this is true, then this triangle is isosceles. Isosceles altitude angle bisector median is the same line. Which If it is missing, you complete it. If Y and A exist You add the K. If there is K and A, you choose Y add. If there is Y and K, add A. So if there are two of these three letters There is a third one and it is a triangle isosceles. You immediately note it. Height angle bisector side bisector If at least two of these triangles are the same line, then these two triangles are the same line. is the edge. Height bisector if two of the medians are the same line The third one is also true. This too immediately You are taking notes. What happened naturally? Even if the question doesn't say the height you you place it and the triangle is isosceles you say. Come here. Now the height and There is a median. Then the problem is He doesn't even need to say it. Angle bisector You show it. The triangle is isosceles You say and that's how I did it here. Let me add the last one here. So what what's left behind? Angle bisector and side bisector. Look, if the angle bisector and the side bisector are the same line Oh, we did that, sorry, I'm correcting it. Finally, what else is there? Angle bisector and there is height. Angle bisector and altitude if the same is true then the problem is He doesn't even need to say it. median You add and say the triangle is isosceles. At work This is the case. So what about those three points? will we write? Triangle above. Above triangles are definitely isosceles triangles. You can write this down as a note. The triangles above are definitely isosceles It is a triangle. In this way, we have important information Friends. Even We have one more important image. This is also regardless of the subject of the questions in both quadrilaterals and circles future. Even analytics is the future Let me tell you. A shape that isn't even a triangle We will connect it to the isosceles, it is important for. Because isosceles is very important Friends. The base angles are the same. Look height symmetry axis. Both angle bisectors both the side bisector and the angle bisector eee are identical truths. Naturally If you look at the image here, from 0 to one Let me show you more. Such a vertical to the base came down and this upright is also two straight lines he cut it into pieces. The right part. This its name is the middle pillar. Let me tell you that. Specifically, friends, the name of this middle post. What is the purpose of phosphorescence? benefit? That this exists. This is actually A and B, if we call it AB the height of the side? Yes. EU median of the edge? Yes. If the height and median are the same line clear information. Of course you can do this in triangles You complete it and this triangle becomes isosceles. Even this green line is also becomes the angle bisector. This is it. For that reason Friends, finish this here. This net information in the triangle is isosceles. This Therefore, such an angle bisector also exists. What is the shape on the right? Look there, let me show you. to you. Sometimes there will be serious questions like this. Just think about the angle bisector and the altitude. Look, an angle bisector is also perpendicular. The question doesn't even need to be said. A lot This is a special situation. When is it available? When I was a twin There is only. There is also your spouse. Equilateral It already has a special isosceles status. Naturally, you will complete this. You will create an isosceles shape like this. you will show these and the edges too you will show that it is the same. For that reason as I noted here, you know this You don't need to write. At least medium when we see a perpendicular or an angle bisector if it is perpendicular to the base, it is a two-sided triangle we can complete. Complete it for sure I'm not saying that but it's a very big possibility. when there is a question of length or even angle It may be, it doesn't matter. When we complete because it is isosceles we will benefit from. Standing on this place It was necessary. That's why I stopped. Now I continue on my way. It says how many degrees is x? Important information what is it? Repeat it again immediately. What you say? Now let's connect it to memory. What am I saying because I have provided the proofs? Well, clear information. Median and height Is it? Then he won't ask questions. You are the angle bisector add. Two of them are on the sidelines. I like the red ones I will give. You will add the greens. I am here. There is one triangle. Angle bisector and I gave the height. The question is telling There's no need. You add the median. Triangle isosceles too. I gave one triangle. I went and said side middle. Let me do it like this. What did we show there? Yes. Median I said. I said angle bisector. The problem He doesn't even need to say it. Add your stitches and the triangle is isosceles too. In this way Let me say this too. I like the red ones I will give. You add the greens to the question You will say that the triangle is isosceles. And also If there is a middle post, you will go like this you will complete the triangle. Two-sided triangle you will say. If you are a half-baked angle bisector like this house. Then here If it is 90°ce, the question does not need to say without having to shape it like this you will complete. Let's see then. Exactly I have what I want. Sees the middle pillar Are you? Middle post. What to do then? The first thing that comes to mind right away is isosceles is completed. When you complete the isosceles Of course, let me spell it out. AB and AC were equal to each other. Beautiful. Here Did you see? Actually, what is spontaneous? output? If I say T here with AC, what happens here? happened? CT also came out equal. It became isosceles. What were we saying? Isosceles The base angles of the triangle are the same. What else? we say? Well, it's 38, sorry. Two inside external. Look at the intertwined isosceles Don't forget to do it too. It was 76, two inside outside. Then the base angles came to the same spontaneously. This is spontaneous became isosceles. Look at BA and AC I found the isosceles equal. Your own question because it gives possible equations We handled the incident this way. 76 76 more 152 So if it's 152, what's going on here? Well, 38° isn't it? Is it 38? No, sorry, 152 It's happening. I'm correcting it. It's 28. 38 28 is 66 degrees Friends. Here is the correct answer to our question Look at it this way, we saw the image. And also What do I want you to pay attention to here? Sometimes such irrelevant equations are given. So what's the connection, teacher? A BTC so there connect and get a new mate from you may want you to find equal sides. Let's come here. No angles are given at all. Well, what is it? There is a middle post. And each other I gave 3 equal pieces. Then I the moment I see the median perpendicular I will continue. For us, friends This middle pillar is a kind of isosceles is the reason. So I'll definitely give it a try this much. I tried. Then I found x. Ok. I wrote the base angle as X. Then we went down to the bottom again. The base is in two divided. That is, the middle pillar. In that case I'm showing. Pay attention to this immediately Please. If it looks like an S, isosceles this is the same. Ok. Then to S The similar image is isosceles, this is also same. Did you notice? What happened? Here an equilateral triangle appeared. Equilateral triangle became 60 60 thanks to its emergence. 2 inside one What does 2x equal from here on the outside? To 60. X what happened from here? It was 30°. Look in this way, about the importance of the middle pillar It was a good question. Yes. Look Let's look at this question now. A triangle is given. There is an angle bisector and a side bisector. There are 55 25. I'm asking what is AGC? to you. What can I say? It looks like this Let's examine. Now when I examine What do I notice there? Angle bisector and median is the same line. One minute. Where is my teacher? See important information first you should do it. Is this the angle bisector? Yes. Same median in time? Yes. Angle bisector and If the median is the same, you immediately add height. I added. The same is true and the triangle is isosceles. So where is equal? each other, teacher? E is the green base We went down to the base accordingly. base in two dividing. So our other edges so this edge that I painted red and This edge that I painted red became equal. We went down to the base. Divides the base into two. It is the angle bisector. Then the base on the other two sides opposite are isosceles. So it would be 55. Look at the base angle and pay attention. The sole is green realize that you are. In this way, it becomes 55 What happened to you? 30° to the link angle. Question What does he want from me? AGC 30 55 more 85 two The soul became 85° from inside to outside the answer to our question. Now let's come to 27. to our question. What did we say here? What is X? degree? Well, X is actually like this We can ask this question more easily. You know, what happened at that moment when you were thinking about this and that? I thought, uh, I did something but at least I wanted to at least show this again obviously. Look, there is a hidden isosceles in the question. There is. Even though it is not very useful for us, Let me tell you. Look, we normally find x. with the help of two inner and outer but I still Let me give you the extra. Let me extend it a little bit solution to the problem. Let's do fantasy. Do you see the vertical line going down to the tab? Same angle bisector in time. Then immediately I'm showing. Here is the edge here that I painted green The edge I painted green is the same. To the base if the angle bisector is clear information is the median. Clear information triangle isosceles. What happened in this way? 3x happened. So let me show it like this. What do I say next? Two inside, one outside Actually look back now you can do. Please pay attention to this triangle meat. Here, if we call the dotted angle y, Isn't the sum of x equal to 3x? Of course not What happened to time? 2x then 2x 2x 2x Look The question is over, so what happens now? 2x 3x more 5x interior angles in this triangle 5x = 90 x what is 18 Of course, this question is answered quickly a question we can handle but there I also wanted to show an isosceles since I gave it to you right Try this for yourself right now - If b 2c, what is the value of x? I ask, yes, what will you say? You may say now What does the red border army have to do with it? What does height have to do with it? What does the angle bisector have to do with it? If the angle bisector and the altitude are the same line Even if it is not a triangle, I will complete it to a triangle. HE such an important shape. What does it mean I want? Take a look. Thoroughly I'm expanding. Let me show it in green. This green angle bisector? Yes. Green base does it descend steeply? Yes. Then it is unquestionable I complete this into a triangle. I'm done. The angle bisector height is also is the median. It is C. Since C is isosceles. Also this Let's not forget. Of course I phosphoresced red edge and red that I highlight edge. We went down to the base. At the same time Since it is an angle bisector, it is a side bisector. is isosceles for . Naturally twins Since it is an edge, it becomes side e. All of them since it is a since it is all a I will show it like this. That one I painted what's coming to the edge? A-B is coming. Question What gave me A - B? Take a look. 2C gave. Naturally 2C. The length of this whole place is already 2C. Look CC is still 2C. The top angle became 25. Then what does he want from me? He wants X. 2 Don't forget to do any exterior. 25 + 25 50° 50 90 x also look here 40 came. This It's not like there's an isosceles figure there. It seems but the angle bisector height is ours very valuable for you. This is how I said it Look at our previous shape quickly let's go back one. Where was he? That one As you can see, the angle bisector height median is the same straight line. Triangle Even if it doesn't work, I'll complete it myself. based on its importance. Now what did our question say? AB and BC are equal to each other. Black color Line segments of the same color except If they are parallel to each other, what is Y? We ask what the degree is. Yes. Now same color line except black parts parallel. Yes. Then a times the blues are parallel to each other. Okay. Then the red ones are parallel to each other. So how do you get from there to Y? can I do it? Let's think about it. Afterwards What does the equality of AB and BC have to do with it? That too Let me show you with colors. With the EU What does the equality of BC have to do with this? Now, have you thought about this, friends? I'm telling you now. This is because, you know nice commentary question. Yes. There What does equality have to do with it? Why the parallelism, sir? In parallelism Friends, let me tell you a good information. We will use it later too. If there are parallels, take note of it. There is a parallel. Is it ok? Arrow like this I showed. I am the one asking the red ones I gave it. I also gave the opener. Important with the star We will use a lot of information. Because of Z, this is also the striped angle alpha. alpha triangle became isosceles. Did you see? Here's the thing. Just because there is that I already put the question. Because of Bak D here the lined angle will be the same. Triangle twin edge comes. The edges I phosphorescent are the same isosceles triangle because it is If I give it, look, normally it is 3 isosceles If I don't say triangle, it's the altitude bisector must give at least two of the medians. None if the triangle is isosceles then we can argue closed. Clearly mark the median why? Of course it is height. Median What exactly is time? is the angle bisector. So what happened now? Look, I had an image like this here. 90 I achieved the degree. Now the parallelism Let's use it. Never seen the parallelism of the blues I didn't use it. I'm extending it. What do you mean when you stretch it? I see? Well, the median that goes down to the bottom it was also the height. Of course What happened to X because of Z? It's 90°. Now that X is 90°, X - 20 The angle here will be 70°. Companion The angles are equal, of course, 70°. Now how do I find another Y? From Z because. My voice is gone. Y e = here due to Z It's 70° now, friends. Our question answer. We've come to another important part. Friends. It's always one of the important parts. We will continue like this, so to speak I told you I would break it into pieces. At work A new piece of ours is the question asked by ÖSYM our identical shapes from the models, our conjugate shapes. By taking advantage of these Well, we will also make angle comments. Here I already wrote it as a note. ÖSYM is identical uses shapes frequently. Identical shape angle and side used in questions We need to pay attention to equality. Spouse shapes, everything of identical shapes are the same. Both the angle and the side length. Now in these same ways, definitely We need to pay attention. Sometimes rotation we will do it, sometimes we will fold it but we use rotations and pairs opposite the same edges in the shapes angles are the same. Just because of this I showed. Look at the equal sides attention. With the hypotenuse of this triangle, The hypotenuse of the triangle is of course the same. Natural Two sker triangles are formed. So you're right here right now you will say. Let me spell it out. EU isn't that the hypotenuse? Yes. BC hypotenuse isn't it? Yes. Triangles are congruent What do I pay attention to when it happens? Of course Since the edges are the same, this is also hypotenuses are identical. Two edge. Since AB and BC are the same, 55 55° happened. Naturally, what remains here? 70°. So the opposite of the green edge is 70. So then this would also be 70. If necessary, here are 20 If necessary, this place is 35 that it is, that this place is 20 etc. I use it. So in conclusion the questions He will hide it inside and not tell me. of course he will say the same shape or He will give the image and wait for you. Therefore, the information given there We will definitely check it out. In that case This is all our knowledge. In the light of this knowledge Let's look at our questions. Deal with it now with this please. What will you say? Below are the peaks in red two identical isosceles triangles specified each side is on the same line. Accordingly, I asked how many degrees x is. Yes. Now the peaks are red since it has edges, it is a top the dots are red dots According to them, their edges are also the same Let me show you what happened. After all, the hill What was the intersection of the points? Equal intersection of edges. Naturally what happened? The ones I show in green are the base happened. Are you at home right now taking the exam? at least one or two such for not entering It might make sense to use color. Well or here's a black pencil, a and maybe a red pen next to it If blue was added, the questions would be a little clearer at first. can contribute to making it better. Look here too, the difference is when you show it like this Did you take the image? An isosceles triangle appeared. If I call an angle like this alpha, this angle I'll say alpha. Look again, my point is this my teacher. We ask about X. What's the deal with alphas? So how do I find x? There is nothing about it. But what is given here What did I say when I used it? Here too identical shapes as in the note equal angles, equal sides We will pay attention. Intertwined In isosceles, make two insides and one outside I said. Then look, there are 2 alphas. Later The base angles are the same, so 2 alphas. And pay attention to this. If the shapes are identical, the base What happens naturally if I'm hungry 2 alphas? The base angle of the other one is 2 alpha. Beware of. So what happens now? Two as you can see, there is an internal and external 3 It became alpha. In that little red triangle I said 2αfa alpha and 3αfa. 3αfa + 102 = 180°. Because it's a linear image. Look What is this place 108 180 - 102 doing? 78 3αfa What happened to 78e alpha? 26 alpha 26a is now 2 multiply by 52° again 52° 52 It became 104. X means from here 76 degree. Look, it was such a good question. Yes, let's come to this question. Below 5 pairs that are identical except for their colors the vertices of the triangle are colored black shown. According to this, what is X? degree? This is actually a thing Inspired by the question, I wrote it here Friends. Well, because ÖSYM is also good He had one question. The questions that came out at least you will see it anyway instead of directly putting the extreme similar It's a question we'll ask by taking a decision from there. You may not even remember, but you will see. Now these triangles are identical and they have a common corner. Look what you see What kind of contribution would friends make? The exact angle we learned in direct angles I will use 360 degrees in this question. How do you mean, teacher? These are all identical Since it is a triangle, the base angles are Aren't they all the same? Look, this is ours. important for. Sometimes ÖSYM asks this in the questions He's trying to do very well. He does it too most of the time, at least in geometry by taking advantage of the identity of the shapes does not give away many things. You notice wants. I don't care about x or anything. I'm not even interested. First, the one given take care. Look how beautiful it is. 5 alpha is here Isn't it 360°? Yes. So then this is each of the identical isosceles triangles How many degrees does the base angle reach? 72 degree has arrived. Alpha here is 72. TO Since the base is an isosceles triangle angles are the same. Look, it's 72°. 72 72 what what's left behind? 36. But my teacher still x We didn't find it. Of course we can. Don't worry. Since they have identical triangles, they are green with the edge I painted green of course the edge I painted is the same. Because the hill dots black dots. Naturally What happened? Our other sides are also equal edges. In this way, an isosceles triangle was born. If the top angle is 72, it is green Let's make the base with the edges. Like this Let me show you in black. It's 108, right? If we subtract 72 from 180, divide by 2 and get 54. O time this is 54. Total is 54. So x what's left? 54 - 18° left from 36. This is it a new isosceles triangle in the figure to see the existence of green edges here for us to see that it is equal was important. So look friends I reserved a special place here based on its importance. Now you do this again You are dealing with. Isosceles triangle perpendicular to the paper base in the form of cut along as in figure 1 and divided into two is falling apart. The parts are shown in Figure 2. a rectangle with sides like overlapping the edges is being placed. According to this, what is x? degree? Yes. Get to work on it now. Here again, an isosceles triangle if we separate it along the direction perpendicular to the base what happens? As you can see once this the edges are the same in this figure. The edges more precisely, the base because it comes the same It is divided into two, sorry, the edges are already same. Now we will pay attention immediately. This How did I place it? Since it is a rectangle, this is 90° Let's not forget that it is. Well, 90° again Since it is, what is the difference here? you need to do? This is an alpha gangbang. This is such a theta narrow place. Huh. HE time is actually here in this triangle the edge seen by the direct height where did it happen? Look like this. So the settlement where is it? Could this be 90°? Narrow Can the angle also be 90°? No. So this means it's 90 degree. At that time, this place was 90° degrees. In this way, opposite 90 this hypotenuse, look here too I'm showing. 90's in red Look here again, the hypotenuse opposite placed like this. So what happened, teacher? A beautiful thing happened. Isosceles triangle was born. Look, do you see? Here it is opposite side 90. This edge here Opposite of 90. Well these are already 90's Our opposite sides are equal. This thus an isosceles triangle was formed. Look Let me show this with color. Isosceles triangle red sides are the same. Isosceles The base angles of the triangle are the same. Peak already 90 because of the rectangle. What came? 45°. What came? 45°. Naturally 45 15 60. Since it is all 90, it is a rectangle From here, x is 30°. Look It was such a good question. Exam You will also see such letters in your questions. There are unused questions. For that reason the view of the edges there for us very valuable. Look now this question another special type after Our part is an equilateral triangle. Equilateral triangle is a special isosceles triangle. Because all the properties of an isosceles triangle has. But now there is a third extra the edge is also the same. So what to say I want? As in an isosceles triangle We went down to the bottom again. The base is in two dividing. Also the angle bisector. Look This is isosceles but equilateral more specifically, all of these are now since all the edges are the same look at the edges, descending here too is the perpendicular bisector and the height. And again this side edge here going down the vertical again are the angle bisector and the height. Therefore the difference from the twin edges extends to all sides the uprights are also the axis of symmetry It's happening. And of course, our must-have All angles are 60 degrees. Edges identical to each other. Come on now, just this Let's look at our specifically related questions. Our first problem is the equilateral triangle ABC. To ABBD equal. How many degrees is DBC 20° CAD? I'm asking. Let's see what we say now Here. ABC is an equilateral triangle. An equilateral triangle times the edges are the same. It's a Let's show. First side, second side. Because friends, if in an equilateral triangle If there is no comment on height etc. in the question If there is no sewing, what will I use? The sides are the same, the angles are the same to be. Okay, I did these. An extra information. AB is also equal to BD. Did you see? A new isosceles is born. Which one? New Where is the isosceles? Let me lift this up see better. Let me raise this one too see better. Look, this was not in the question. By using the equilateral equation, we also have BD a new one, taking advantage of equality isosceles triangle arrived. Then immediately I'm checking. Because it's 60 here 40 left. Because it is an isosceles triangle Subtract 40 from 180, 140. Divide by 2, 70, 70 happened. What do I want? CAD Where to get CAD I find? Since it is an equilateral triangle, 60 70 - Look, it has become 10° from 60. The answer to our question This is a simple question. AC is actually 33 I forgot that I was going to replace 34. We solved this first. Easy now We are solving. It was no problem, you know? Let me tell you. Equilateral triangles ABC and AD If so, what is X? The move I will make What? Since ADE is an equilateral triangle, once Subtract 15 from 60. 45°. What else can I say? my teacher? Just this equilateral triangle the edges are the same. 1. Equilateral triangle angles are also 60 degrees. That's why Just type C as 60. Here it is The question was as simple as that. 45 60 more 2 inside What came from outside X? 105. Just look at that he saw this. So why this question? you are asking, teacher? Equilateral triangle Please make sure the sides are equal, Make sure the angles are equal. Let's look at this question now. In our 35th question what do we say? A square with an equilateral triangle a triangle with two corners on top of each other over the edges What is the value of x when placed? Yes. Square and equilateral triangle. How in between will we establish a relationship? Let's think. What do we say? Once upon a time, friends What is the property of a square? Corners 90 degrees degrees and sides are the same. The edges are identical. Equilateral triangle. Its edges are also opposite each other the same. So sir, once the square Another feature is the mutual the sides are parallel. 90° is 90° Look for the information I have given you. Could you please reconsider? Karenin the edges are the same. Sides of the square parallel. The sides of an equilateral triangle are the same. Think now. He knows what our topic was Are you? Let's put a star on this. Question type because it will be useful for us after that because of the square If there is parallelism then it is 60°ce of course, 60° again. 60° is still 60°. So what happened with that little red triangle? It became an equilateral triangle, friends. This What is its contribution to us? Color it again immediately I'm showing. Let me do it like this. Look, The red edge is one side of the square. Same also a side of an equilateral triangle. The red edge is one side of the square. As you can see, let me remove this now. from the middle. The red edges came the same. And here is our triangle Let me write A. Look. Our triangle B C is a became an isosceles triangle. Letters I'm lifting. Now the triangle is isosceles When I see that it is a triangle, the top angle is something one corner is already 90°. From an equilateral triangle because it was already 60. 90 60 150 base What will be the angles? 180 - 150 30 Divide by 2 It was 15 degrees. So now it's 15 60 What will remain in the triangle with our 3 angles? 105° the sum of the interior angles of the triangle that will remain because. That's our topic, friends. These are the basic outline shapes these. At the same time ABC is an equilateral triangle. Equal to ECD. USA 40° What is the measurement of DAC in degrees? Control Let's. Now EC and CD are equal to each other. What good will it do me? Let's see, Let's think. What is it once? ABC is equilateral Since it is a triangle, just like this 20 Let's put the degree here. Edge can do something extra for equality Am I? Why the slight equality here? given? I'm thinking about these. Look Normally I can't say anything. Where is the equilateral? the equality of the sides of a triangle It didn't work. Well there is a new isosceles There is. What does that have to do with it? Let's take a look now. Once this angle can already be found angle. If you ask why, sir, an equilateral triangle Oh, ABC is 60 degrees. don't forget. What happens naturally in 60 40? So 100. What's left? 80. So look the sides of this isosceles triangle are e Sorry, I actually knew the angles. I was finding it. Did it become 80 because it was 80? again? Yes. Then there are 20 left here? Yes. Okay. I still don't understand, teacher. So what's going on here? Let's take a break if you want. You too, stop. One upon one think. I need to think here. What good will these do me? Base angles of an isosceles triangle Finding helped me find the angles. After that, yes. Here is this thin I wrote the question so you can see the part. Look 20 80 is a beautiful triangle. Because look here Please be careful. What does 80° and 20° do? 100. What remains? 80. So Friends, angle C is also 80. Difference Did you? What will happen, teacher? Right away again Let me show you with color. Look Side BD is equal to side BC. Because base angles were 80°. BCD ikar was detected. This is also due to the equilateral triangle E greenside. Did you notice now? New an isosceles was born. An equilateral triangle new duties when appropriate is to find isosceles. Different by showing the truth. What happens next? happened then? We saw this. Isosceles triangle. I'm deleting this. He did his duty Now. He saw us in twos. He made me see. What what about? It made us see. This is how we did it. We saw two yellows. New others because an isosceles was born I'm deleting it now. I'm done with this I say there is no more. Don't find this place completely This was his job. Now the new isosceles Our triangle also has a vertex angle of course Since we know, what will happen to 40°? Base Our angles are 70 to 70 DAC in this way If we subtract 60 from 70, this little thing here 10° remained to our angle. Look like this It was a good and important question. New noticing the equality in both this question and This question was already very important Friends. The two are equilateral. Here's to this It works. And sometimes friends ask you questions It doesn't even give a triangle. If the angle between them is 60 degrees, what is it normally? I wish I knew what to comment on. but the angle between them is 60° and these lengths If it is the same, I will give you a clear message and combine it. means. E combined to form isosceles base angles are necessarily the same as in the interpretation It's happening. A new equilateral triangle is born. So if the ikar around 60 is equal make us find a new equilateral triangle is for. Let's see for example right here What I mean is, what's in our 37th question? I call something isosceles isosceles. TO once there are 60. It wasn't enough. And also 60's These two sides around it are equal. In that case Clear information tells you to combine this. When we combine, our triangle is now equilateral Since it is a triangle, the entire angle is 60°. All of them 60°. By the way, because of 100 degrees The base angles are the same anyway. This isosceles 40 40 of the triangle. Since I found the whole 60 What is left for x now? 60 - 40 to 20° remained friends. Yes. You try it right away at 38. Before You take a look. Now let me enlarge it like this also. In DE, EÇ and BC are identical. 110° He asks me for the AED angle. What you should see Is this question necessary again? 60's if the two edges around it are the same, immediately combine this. I combined it. When you put them together, you see that they are equilateral. 60 and 60. What's left? 50°. Then the base angles turned out the same. Wait a minute, teacher. How? Look, double lines. Double line Do you see? What happened in this way? Now Because it is 50°, 50° 50 100. What is this place? happened? It was 80°. So now Thanks to this isosceles triangle, friends. 80 60 more 140 here left 40°. This is the correct answer to our question We found it. Yes, in an equilateral triangle I told you before, but Let me say it again. The entire equilateral triangle sews the heights to the base. It becomes a side bisector and an angle bisector. And now our new information. In this way, all The heights are the same, friends. All heights are both angle bisectors and the same median in time. Even divided Even the parts are the same. Be careful Let me spell it out. AO with OC e= OC with OB is equal to. Let me write it here too. OC, OB, OA all are equal. Let me even write it like this. O, then OT and then or. Here Friends, all of these lengths are identical to each other. I showed it this way. Then look at the angle bisector 30 degrees 30 degrees 30 degrees happened. The 90s were steep. Look at all of them as you can see. And the dotted angle look thing dotted edge none of the symbols are the same not different. TB here is TC then CSA then ar and then RB. All of these lengths of friends the same. Because the triangle is an equilateral triangle and the edges are all the same It's happening. The parts are also the same. Natural As the heights are equal, we will be very valuable for you. This question right away Let me show you through. Look ABC is equilateral Since it is a triangle, how many degrees is X? I asked. Let's see what we can say. Now Our topic is this. I'm giving the tip here. Friends in equilateral triangle questions gives you one height. Is it ok? Sometimes there are patterns like this I will tell you. Gives one height Actually. Draw the other height so that two unrelated scalar triangles appear and so that we can handle it. So look Normally, please pay attention and say ah. here. Ah la'la let me say R here. Let me call it S here. What does AH have to do with RS? isn't it? Identical edges. But what do I know? The heights of an equilateral triangle are the same. No middle in an equilateral triangle is given to you in vain no points are given. One more tip is coming. The second feather is in an equilateral triangle Why does it give you the middle ground? Midpoint very special. It's true that you downloaded it here if it comes out from the corner, it is a perpendicular and an angle bisector. In that case His message to you is this. I am equilateral to you If I give the midpoint of the side in the triangle I am giving it as a note and draw the height. Therefore, the midpoint of the edge is I give a point P so that you can use DP draw. Look, this is important. You noted it, didn't you? If it gives the midpoint of the edge waiting for you to draw the height. Natural What were the heights like? Equilateral identical in triangle. Then BP too Let me make it green, oh let me make it green. Equal. By the way, where is AH equal to? was given? The question gave itself. PS difference Did you do it now? A new one by itself isosceles triangle was born. PP and S equality also occurred. From where? Because In the question, I gave AH and PS equally. Naturally this isosceles triangle What will we see now thanks to? Isosceles triangle. Well, if I say alpha here I would also call him alpha, but I don't even have to call him alpha. no. What do I say? In an equilateral triangle The altitude is also the angle bisector. 30 It's 30. Since the base angles are the same 30°. The edge that goes down to the bottom is also is high altitude. Be careful. This So now 30° 30° what does it do? 60. What's left? 120. What does it take to be 120? happened? See red angle 30°. The one who completes, since it is all 180°, it completes Our angle is x, what did we get from here? 150°. At work Look at the equations in this question we should have used it. We made one, it's not enough. Let's do another one. Come on, this and that You deal with it. Let's see. Both What did I say? ABC is an equilateral triangle. AK = HC. I gave height. Heh. Same thing again We are on the ground. Look, here's our topic. So sometimes it's like this, you know, it's kind of like that to apply the memorizations in quotation marks not bad. So I'm on a new path I will produce. No, what's the need? Point D that's the situation now. Is point D the midpoint? in an equilateral triangle? Yes. Why the middle point? gives it to you? Draw height. In that case I drew AD. It's the same with color Let me show you. Meanwhile, the equilateral triangle be careful heights are different from each other is the same. That's why AD and HC are the same I made color. In the question, HC and AK are equal. I also made AK green because it gives. Now what happened to our new image? A new isosceles is born. Because this Look, I painted the green red. Green I painted it red. These are equal to each other came. I'll even make them all red Now. became an isosceles triangle. Then this did his job. I'm deleting HC. A new We obtained the isosceles triangle ADK. What what will we do next? Equilateral Since it is a triangle, oh the height Remember that it is an angle bisector. 30° 30° After that I gave this 80. In that case If 30 is subtracted from 80, two inside and one outside are left. It remains again, look, there are 50° left to D. 50° since it is an isosceles triangle N was also 50°. 50 50 100 back 80 What happens to x so that it remains? 50°. Inside the triangle Look at the sum of the angles and find x to be 50° I became. This is it, friends. Irrelevant the equations that appear to be equilateral triangles due to their same height we use. Yes, let's see. Now the green edge of the upper equilateral triangle isosceles with a red vertex one of the sides of the triangle is parallel. Accordingly, x is expressed as We say what is the measure of the angle in degrees. Yes. Now once isosceles the vertex of the triangle. Ok. In that case What should you pay attention to immediately in an isosceles triangle? will I? That side of the equilateral triangle is Who will time be parallel to? Of course It's going to be parallel to this edge right here. Let's show you. Let's do it here. E color Let me do it like this. This is the lilac purple that I drew Anyway, these two edges are connected to each other parallel. Later equilateral Of course, it is due to the triangle, like this Let's not forget a 60 degree. In that case 60° due to Z. So in the question Z It was important for us to see the rule. What will you say next? Here in the little triangle I colored it red two inside and one outside in a triangle equal 100 What was the apex angle for ? It's 40°. The sum of 60 and 40 is equal to the exterior angle. Now it was an isosceles triangle. Of course 40 out of 180. 140 divided by 2 70 70 My base angles are 70. What do I say next? Be careful here about the equilateral triangle. due to 60 and what came out here? one m rule. Look, let's get that out of the way too. What happens if the angle is 70 and 60 on a straight line? It is equal to the outside angle. M over there As you can see the rule is like this I've shown it. This way it is beautiful was the question. Let's try this right now. Below the identical arm scissors shown in Figure 1 one of its arms rotates counterclockwise x is rotated by an angle. Accordingly, y how many degrees is it? Counterclockwise x We rotate it by an angle. Well then, look I say immediately, of course, it's an effort. Like this I grew up. It was normally like this. Hour How much did I rotate it counterclockwise? X I rotated it by degrees. Then this is it friends Our angle is x°. Like this, with arrows and lines Let's show. After all, their arms were identical. Their arms are identical, meaning they are the same length. The lengths I drew in red are the same. HE What happened when I turned back time? This The angle between them is x degrees. E friends I have already given x here. How is it useful to me? Then let's check it out. This is what I did now. H x 60x. Ok. Did you notice it here? A new isosceles triangle appeared. I think that's important It is possible. After all, when I turn that arm equals here. Well, the red ones are already equal. Ok. So then e is x We can write an angle using . How do we write? Let's see that too. Normally Y says here, I shouldn't call it Y but Z You say, you say Z, but I don't think that's it. no need for conversation. Subtract from 180. What what's left behind? 120 - x. Now divide by 2. 60 - x/ 2. Look how beautiful it is. Here 60 - x/2 I wrote. Since the base angles are the same I wrote 60 - x/2 again. I'm telling you right now. This is because the arms are the same again side, this side and this side are the same. HE If I letter the time, corner A, B and C When I say AB is equal to BC, isosceles The triangle was born. Is it ok? Now this is it What will we say after seeing it? Well 60 - x/2 follow. Red edge thing red angle and here is x. As you can see, C What is the sum of the angles? 60 - x/ 2 + Friends, we got 60 + x/2 from x. And red edges are identical because it is even green now I will paint it so you can see. Look at the BD edge If the side BC is the same, what is the angle Y? came? That came to 60 + X/2. Open the hill how many degrees? X°. Then X + 60 + X/2 + 60 + X/2 Interior angles in triangle BDC total 180°. Thus 60 + 2x = 180. x What happened from here? Well, excuse me, X is from here. what came? It's 60°, right? 120. Yes X A wait a minute X why is it 60? We wrote it wrong. I will write 60 60 more 120 I wrote 60. Excuse me. 120. Then 2X from here is 60. Yes now We found it right. X = 30 arrived. There was a moment I forgot about that 60. I fixed it. 60 120 I did. So now x is 30 The answer already requires Y from me. Immediately I show the triangle again. Well, the hill My angle is already 30°. Isosceles triangle one of the angles is Y and the other is Y. 180 - 30 150 Divide by 2. This is our correct answer It's 75°, friends. This is the thing. We'll do this later with the circle interpretation, but Now our friends who don't know hoops It's happening. So let's put this in the circle. Let me. In the circle it is actually equal a center B thanks to the edges circle and from there the central angle the circumferential angle We will comment. Don't worry. Let's come here now. Look, I said it like this or I didn't like it. For those who don't know, nothing no problem. Look, I already solved the question. It's no problem for those who don't know. Circle Those who don't know aren't making any fuss right now. For our students who know hoops I'm telling you. The rest is immediately on to the next question can pass to deal with. From one point indicates at least three correct circles Friends. If they are equal to each other. And these already equal. Then center B is as follows we can draw a circular arc. Is it ok? Naturally, due to the central angle, what happened? It's 60°. What you see from the surrounding angle is half. What happened to x because it reached 60? Half of 60 is 30. I rotated it by x degrees. For the top angle is 30. Look at the rest same. This is how it is for those who know the circle Questions could also be asked in an additional way. Deal with this immediately. Beautiful Our questions have arrived. Look back to back Thanks to the equilateral, there were also equalities. Also, there were three things lying on a table below. identical compasses are given. From compasses The two arms intersect at right angles here Look. And the points A B C D are collinear. Ok. Like this. So let's try to figure out what x is. Now I see a shape like this Let me enlarge it. Both my arms and legs are identical. The identities immediately remind me of isosceles should show. Look, this is how I do it. I'm showing. P Let me say. PD and CD are equal. Isosceles triangle there it is. Let's not forget that. Even Let me show it with color. Oh look. One arm of the compass. One arm of the compass. These are identical. Attention. There is an isosceles. Then because of the compass This is already isosceles. Later when we open one arm of the compass 60 degrees of course the base angles are relative to each other is equal to and 60 is 60. Because compass their arms are the same. Naturally 60 out of 90 what is left because it is? 30°. E B If the angle is 30°, the base angles are the same What will happen because it is? Angle D is also 30° will be. Beautiful. New here is an isosceles We said triangle. This is automatically equal It was because of the edges. 30 out of 180 interest. 150 divided by 2 is 75 and 75. Then Friends, 75 60 is the correct angle of this. If we subtract 60 and 75 from 180, what do we get? remains? 45° remains to x. Here are my friends The questions are simple to solve. Oh teacher, you When I solve it, I understand how beautiful it feels. Look, geometry can be deceptive like this. Where is it? We will not say we understand when you solve it. Well, Oh, you know, that sentence a little bit We need to distance ourselves from it. It's tricky because it's like math not geometry. You will struggle. Even After I answered some questions Try it yourself again if you want. Check out some of the questions in the private lesson. living is normal. Don't think that most of the students are taking the course I immediately asked each question one by one in class. The wheel understands very well. What is he doing? do you know? Where is my own? I remember it from my student years. Another Well, he somehow figured out the math Here is a note for students who are You put it. It becomes a star, a question mark happens, something happens. Then he goes home He deals with this himself and thinks about it. Well, what did the teacher say about this? instead of saying first yourself again tries to understand. Then what the teacher said remember and look at this there was a connection. Because friends The questions are not chosen randomly. So I Write down the questions I like and ask them to your I do not present it to you. All of them are related to each other connected. create a schema in one's mind I'm working on it. When you make an effort too is occurring. Don't worry. Newer We are going. Now we come to our new section. After the isosceles equilateral one wishes can think of this place as a new break. It's already one piece, you know, it's still going on right now I do but I will do these separately later I can also break it into pieces. Well, edit it But here are the ones that left in one piece right now can exercise his/her right to a break if he/she wishes. Others will already see it piece by piece We will go for 20 to 30 minutes. We are in the important part. Right triangle again the triangle we will use on the ground. Just now I'm using angle properties. Look angle in a triangle by dividing it into sub-parts These are the angles in a triangle, Bodoslama said Look, we are not continuing. Each one separately We examined it separately. Isosceles apart We examined. We examined the equilateral separately. Separate use of identical objects We examined. Now we come to the important point to the triangle. Friends angle in a right triangle The feature is this. The magnificent trio. Friends in a right triangle Let me tell you right away, it's a right triangle. why? Its two edges are perpendicular. A and B we call the edges perpendicular to the edges. Also to C we call it the hypotenuse. To the longest side Friends, we call it hypotenuse. Later We will see Pythagoras. Enter the angle and length I'm not entering. And here is something like this We have a feature. Let me tell you that right away. Meanwhile open-ended I will say it again in the context of relations but the largest angle of a right triangle is just a right angle is the angle. The remaining alpha and theta It is definitely an acute angle. A right triangle The largest angle is 90 degrees. The biggest the side opposite the angle is the hypotenuse. What kind of feature do we have? The magnificent trio. You've probably heard of a Let me tell you what it is like. A steep median of a triangle descending to the base the median is also like this friends are equal to parts and The magnificent trio is born. In this way a We have a feature. This for now Let me show you in red. The magnificent trio That's what I'm saying. The median divides into parts equal when but goes down to the hypotenuse. And Friends, we have learned something in geometry now. I'll tell you. Three in geometry If two of the information are given, the third is clear There is. You know where we implemented this? Are you? Isosceles triangle. Height median. What is my third piece of information? Bisector. HE time does not ask questions. I will see. Then friends, what did we say isosceles? in the triangle? Look at it this way. Well, the median, The angle bisector does not tell the height. We we will say. Here's what the situation is do you know? In a right triangle if If he gives me the middle of the line, the magnificent trio I will see. Attach it here now. Look Let me show you with color. You do this you will show. Does not ask questions. Wonderful will not give the threesome. Missing parts will give. We will complete it. 2. Three I gave away two of the pieces. Like this not in the form of a median but like this I gave it. Look, there's definitely a third one too. equals. The question does not give red. You you will show. Again these two on the right if the part is equal, the 3rd on the left You will give the piece. Now the star I'm putting it. The only shape with the magnificent trio is upright It is a triangle. Then there are 3 undeniable truths if he gives us equal parts Our duty is to draw the right triangle there paste. That's why friends Let me tell you the information like this. Strange We will use them to some extent. Well, even Let me show you this too. Look, teacher, why? is this the same here? Look, here's why. Alpha de alpha de. Let's call this theta. Follow-up meat. Isn't alpha + theta 90? In that case Friends, look at this angle equally. Even if I don't know it's alpha + theta 90 90° to the third angle in a grand triangle It has to remain in theta by itself, look isosceles came. That's why it's okay It's not even a thing. One minute Where does it come from, sir? Coming from angles again. Now look here. Where is it? Why is it 90° here, sir? I said alpha. Alpha I said. Let it be an angle I don't know. Theta I said. I said theta. Note the large triangle meat. Isn't 2αfa + 2 theta 180? In that case What happened to theta + alpha? 90. Seriously, what is it? The top angle is 90. So that's the meaning. What you attribute is so problematic It is nothing. Okay sir, okay, they are This is okay. Okay, that's it. This The second and third are the same. I'm over it. TO Because look here again when you say alpha alpha theta pattern 90 alpha in the grand triangle it has to remain theta. Okay then We understand but why is this equal? That's also a Friends, about our next topics proof but let me say it anyway. Middle I'm pulling the base. I'm shooting parallel. Because I drew parallel to the opposite edge parallel counterpart edges at the same rate divides. That's what I did. Parallel reservation What happened here? Look at the corresponding angle of 90° was 90°. And this way, now the base If the descending vertical side is the median, what was this? Same the triangle is isosceles at the same time was saying. Look, here comes the isosceles. Here's the proof, friends. It was easy. Look what happens next we will say? If he gives you two, you give him the third paste. All three of the magnificent 3 gave. He didn't say right triangle. Guarantee We'll call it a right triangle there. Therefore I'm coming to this question. Look, he gave them both AD and DC. Then clear information is in BD equals. I pasted it. Then I look at what gave another? He gave one more piece of information. It says AB is equal to BE. AB is equal to BE. Oh, there's an isosceles triangle here too. Beautiful. If it is 42°ce then 90 42 what what remains? 48° to A. Then what what should I say? Isosceles triangle. Look follow meat. Aren't AD and DB equal? Then 48° happened. Since it is 48°, it is an isosceles triangle. triangle. I delete everything else. Now look at the isosceles triangle. Equal to AB. If the top angle is 48, then 180 is 48 interest. 132. Divide by 2 to get 66. Then 48 + X Since it is 66, I subtracted 48 from 66. X, my red angle here is 18°. Yes. Please deal with this first. AB parallel CD. Well, then two unrelated equality is given. Like this, with letters Let me tell you. Look, BH and CD are together given equally. What's the point of parallelism? And x what does that have to do with anything? Let's see. Strange. Now friends, what should we see first? is it necessary? Of course, whatever the topic no matter what the question is, isosceles we got to the bottom in a triangle. The base is in two divides. You should have shown the H yourself before. After all, look, this is an isosceles triangle. When we got down to the bottom, the bottom was immediately cut in half I showed that it was divided. Oh, nice. A steep in triangle H, the perfect triple angle to the midpoint You can combine this in your questions. I combined it in the name of the magnificent trio and It was a beautiful sight. Equal equal equal. What does this tell me then? Equilateral triangle. So, it's 60° here. Ok. HE time here was 30°. Parallelism if CD is parallel to AB, then What was the parallel there according to the C rule? 90. Then of course angle B is also 90. Sum of angles between parallels It was 180. Because 90 - 30 to 60° A is beautiful. Actually this is not isosceles but hidden equilateral triangle. 60° naturally of course 60°. X is also 30° from here. Look at it this way. We handled it beautifully like this. Let's try this right now. AD, BD, EC are equal to each other. B is perpendicular to AC. ACE 10, ADE 40°. Look, sometimes ÖSYM asks questions here. gives the information. You show it there required. He didn't show the steepness but wrote it. This was a symbol of uprightness. B is perpendicular to A, AC I stated in the question. In that case Friends, the symbol of steepness, by the way I'm in reverse. Then if BA AC is perpendicular, this is actually the case. It was 90 degrees. Since it is 90°, 1 2 He didn't ask a question. I am part 3 I'm adding. The magnificent trio. Spontaneously came an isosceles. Did you see? With E CD was equal. Naturally here now what can we say? This is an interpretation where we will use two triangles we need to do it. Where is angle EB? This is it. We will find this. This is red the angle I painted. What can we do? From this After that, let's get some math skills remains. Because we did everything. Now dealing with transactions a little more easily can make a nice comment on behalf of Are we? Can we improve it? Or Are we going to give letters and do it in bodosta? Look Actually, there is a nice comment like this. I The magnificent trio is now alpha come. Look, we have a rule here. Do you see? In a concave quadrilateral what was it? Where was alpha 40 10 equal to? Here. What came? Alpha + 50 has arrived. Beautiful. HE since the base angles are the same Is alpha + 50? Yes. Even even Please be careful. This whole place Isn't she alpha because of her gorgeous top? Yes. Then what happens? Let's open alpha, One angle is alpha, and the other angle is 90 + here alpha. Let me delete this too. Where is the isosceles? Thanks to you, we now found that alpha + 50. Look at the angle d thanks to the isosceles I also found it. Then the ADC here in a triangle, one angle alpha, one angle alpha, an angle 90 + alpha = 180. What is alpha from here? happened? But it's 30° from me He wanted the ECB. Be careful. Total 30 Since that is the case, what remains here? 20° remains. Look, this is also beautiful. was a question. ABC is a right triangle. Be = EC. We ask how many degrees BC 20° AFD is. Yes. Now that's good. Take a look now How will you do this? Now BCA is 20° with EC equal to . Let me show you with color. Let me take it like this. ALSO. I'll show it in another color. to E equal. Yes. If these two are equal in a right triangle the third is equal. The magnificent trio. Let me make all of these green like this. All three are identical. Ok. We did that too, sir. The magnificent trio we saw. Then let me lift this up now. Please be careful. Descending to the bottom actually divides the vertical base in half here triangle BDC is an isosceles triangle. Him you should have paid attention. So what happened? The base angles are the same. 20° here is 70. Of course which is also the case here, it was 70. AFD where? AFD is our current angle. Ok. Wonderful If angle A is 20 degrees because of 3 Some of them are still 20, right? Of course like that. Well, what's left here now? what else is left? To find the point angle. 70 70 here look 140 back here 40° remained. 20 40 more 60 P angle is 120° remained. As you can see in this figure, the angles by writing one by one but especially this question You ask what the star point is? Thanks to the magnificent trio, these are equally output. And here is the steep slope that goes down to the bottom There is. Then the clear information is BD and DC When you see that it is equal, immediately divide 20 by 20 paste like this. Here too 70 70 sticking was really important. Let's see now this question is also good. ABC and BDC a triangle. AE angle bisector BEED ED equal. USA 100° ACD 120° He asks how many degrees BED is. Yes Let's see what we say now. Which topic? no matter what, no matter what question clear information if the angle bisector is perpendicular to the base first isosceles. Look, that's for sure. There is a magnificent trio of conversations I'm not doing it. I don't approach the question that way. What am I saying? Net information angle bisector base If this triangle is standing upright, it is also isosceles triangle. That is, the vertical that goes down to the base is an angle bisector and a side bisector. Now this Thus, the only triangle in which three are equal is a right triangle The triangle was a magnificent trio. The magnificent trio Thanks to you, I also saw the right triangle. What what should I say? E ACD 120°. All 120°. USA 100°. Yes. What does he want from me? BED angle. Where is BED? Oh, here wants. Ok. Interesting. Now what? what should I say? My first comment would be this. Since there are two triangles, alpha here If I say this, you will naturally call it alpha. Then what happens? Look here is 100 - alpha. Our angle right here is 120 - alpha. BDC The sum of the interior angles in a triangle is 100 - alpha + 120 - alpha + 90° = 180. From here 210 90 more things 220 90 more things 310 - 2αfa 180 130 = 2αfa alpha e= friends, what happened? It was 65°. Now if alpha is 65 If I subtract it from 100, I am left with 35°. Wonderful From the sum of the interior angles of the triangle 3 to 35 I found 110 degrees. As you can see, this We solved the question nicely. This first way. That's what I did. Is it ok? The second way is for those who know, and for those who don't I already did it for you, look, it's completely in the triangle. I solved the question with angles but an extra Let me tell you the way. Sir, the sum of the interior angles of a quadrilateral is 360 degree. Can we do it from there? Yes. How many degrees was it in here? 120. What is this place? degree? 100. E because of magnificent 3 We found 90. 120 100 more 220 90 more 300 10. What's left? 50. Half 25 half is 25. Look again, this is 65. Again isosceles 65. Again 35 and again 35 and again 110. That's it, friends. from the interior angles of the quadrilateral We arrive at the same place. You know, no problem. but the important point in the question is ABC It's also amazing to see its isoscelestial was to see the trio. Yes. You deal with this too. BD to AC I said upright. AE, EB and DC are 3 equal parts. AEC I ask, how many degrees is the angle? AEC where? Yes, that angle. I'm asking this. Yeah, let's see. AE and EB are equal, DC is also equal. There is irrelevant equality. So with DC What does EB have to do with it? What does AE have to do with DC again? Most at least AEB. Ok. Now what? can we say friends? Angle questions, especially about right triangles if it does not draw at the midpoint of its hypotenuse you draw. See more about the magnificent trinity We made the comment. Sometimes in questions He won't even draw the middle point. He will wait for you to draw this place upright when doing angle questions in a triangle, naturally When I draw it as isosceles triangle appeared. Did you notice? Couple line, double line, double line. Already Since he gave the same question, he got double again line. So what is this angle actually? It was 23° degrees. What happened to two inside and one outside? 46°. Base due to isosceles triangle angles angles became equal. 46°. Now Let's look at the triangle. AEC triangle AEC. One of our angles is 46° and the other is 23°. 46 23 what else is he doing? 69. So it's 111°ce Friends, what the question asks of me is AEC angle. Try this right now. Well, x is twice y. Two identical pieces. There are two angle bisectors. How many I ask what the degree is. Yes. What can we say? Let's take a look. Now here once again It's an image I showed you so you can get used to it. If I say alpha, alpha, theta, theta, friends comes across you. 2αfa + 2 theta. Look 180 the intersection of two angle bisectors is necessarily What happened? Two separate ones in one line When you draw the angle bisector, see alpha + theta 90° happened. Right then, with your permission, 2αfa I'm deleting 2 tet. Now here it is like this I scribble and paste some 90. 90 thanks to the first edge piece second piece equal. The magnificent trio. In that case One of the angles is y. Beautiful. TO The other angle with the dot is Y. In this way, What happened? One angle is 2y, one angle is 90. X is already It was 2 years old. 4y + 90 = interior angles of the triangle 180. 4y from here 90 4e If we divide, Y friends, it becomes 22.5. Y 22.5 when it is already 90°, open it up 22.5°. Let me write it like this. What remains? To A It remains 67.5 degrees. This is how we do it We handled it beautifully. Then friends I will give you some information. The result is steep in the triangle sometimes students think gets stuck. To fix that problem, I need to give the information. Yes, the magnificent trio comes from here. Ok. But friends, some of them are 90 degree he thinks. Oh my goodness. Look now a right triangle. An irrelevant angle I will give. These are identical. Let's say it's 40 degrees here. It's 40 degrees here let it be. So, what happened? This is 80°. A magnificent trio indeed. 50 50. You saw Is it? In a right triangle, this base descends The median is not 90°. Like that There are those who think so. But this I also understand a little bit of their thinking. How Do you know there are shapes? Sometimes a It becomes a triangle. Such a magnificent trio It's happening. Well, this is also 90°. How is this? Is it possible, sir? Look here, there is no I've seen it, but I also know images like this I. Here, friends, I wrote the note below. If a triangle is an isosceles right triangle Okay, alpha is 90 degrees. But isosceles If it is not a right triangle, alpha is definitely 90 It is not. As I just did, this is a isosceles right triangle? No. Look 40 50 the angles are different. The edges are opposite each other not the same. A and B are equal to each other not. Then what happens? See alpha 90 It sounds different from degree. When a triangle is isosceles. Yes. In that case What was the height also? Median and the upper bisector was . Our only condition is one The only requirement for the median to be 90° is the perpendicular whether it is a triangle or not isosceles is that it is a triangle. That's why friends 90° here in the magnificent trio questions If you want the triangle to be isosceles must be. Is it ok? He stated it like this Let me be. Sir, the triangle is not isosceles. HE Time is not at 90 degrees here. A special It's not a big deal, but it does get confusing. The only shape where the median is the height what is it? Isosceles triangle. Whether isosceles whether it is perpendicular or normal isosceles difference It doesn't. But if a triangle is isosceles then yes is the median height. His Except, don't call it 90° here. Him Let me state. It was such a warning you. Now ABC is a triangle, BDC is linear, DC is 2AB AD is perpendicular. How many degrees is AC BAD? Memorization question type. You will find it in books I'm telling you it's coming, friends. Where is it? Is it very important? What an annoying question. models may be. That's why we Let's know. By the way, it is also in the MAP book. That's the reason why I gave it. You know, just because it's in test question banks etc. not. ÖSYM, well, I mean ÖSYM. Map asked. Let us ask then. a special is knowledge. Just note it like this get it. I asked the question but first note the figure meat. He gives it to you. Is it ok? He says This is x units. This is 2x units. Here and that's it, sorry, I'm correcting it. Like this He calls this 2x x. All 2x. This is like this, here is like this x. Now Do you know what's coming, friends? Questions about length in a right triangle Normally we immediately hit the magnificent triple bang We don't think so. But x is 2x special is the situation. The magnificent trio myself If I pull x x did you see? with the EU edge ac side became xx. It was also twin-edged. HE Look at the time, if I say alpha, if I say alpha, it's like this. a 2αfa base angles equal 2αfa. This Make a note of it immediately. Is it ok? When? To 2a If one of them is twice the other, then a Let's say the hypotenuse of the right triangle is 2a. 2 times a side. Immediately memorize You make the perfect threesome. You did it. Oh, oh you say and look, let me do it here too. HORSE Here the EU turned out to be equal. Isosceles Since the T angle is 40°. Natural It was 20° 20°. Where is the BAD angle? Oh, he's asking about here. It was 90 at that time 20 for came here became 70. If I subtract 40 from 70, drink 2 and look outside. It was 30°. Here is a memorized question like this type friends. Come on, one more Let's see. If y = 2x, what is alpha? I said. Let's check it out. y = 2x. There is a magnificent trio. X x. I placed these. Then y = 2x. I see then it is 90°. Ok. Then this x, this is 2x. What I do? The magnificent trio I memorize it. Memorization question type Because. X. Then here is the right triangle for here is x and again like this red x on the side. As you can see the red edge x. He had already given an x. isosceles came. Alpha alpha 2 internal an external 2αfa happened. Then I called it 2αfa here too. I placed the angles like this. Now There is no different angle than the previous question, sir. HE You will look at the question a little more later. Maybe you looked, I didn't see it, I don't know, but Here I am again, one inside and one outside I caught it. Alpha. There's alpha right there too. Look, let me just delete these. I saw it. We've already seen what we needed to see. Is it ok? I found 2 alphas here too. Even this Let me delete it with your permission. The purpose of the event to keep you focused. Look, there are 2 alphas. alpha. 2 inside and one outside due to 2 inside and one outside Shouldn't this be 3 alpha? Yes. Since the base angles are the same, xx is again 3 wouldn't that be alpha? Yes. Well now at least as you can see here I saw that this place also has 3 alphas. Now Since the entire angle is 90, what is left for this angle? 90 - 3 alpha. Just now the isosceles was magnificent It was because of 3. Don't forget that place. One See step by step to solve the problem. We got pretty close. On the one hand I check the time. Now this is it 90 things, let me delete this, sorry. 90 - 3αfa thanks to this, yes, 90 - I know it's 3αfa. A little more Let me enlarge it. Let's focus on the event. 90 - 3αfa and inside 2 inside and one outside what did we say? Intertwined When isosceles triangles pass, look at 90 - 3αfa 2 more times, let me write this properly. 90 - Where else is 3αfa equal to 2αfa? What's here? came here? 90 - alpha has arrived. Here What? Again 90 - alpha. Then this is it as you can see in the triangle 90 x 3 270 - alpha - what happened to alpha in total? -5αfa = To 180. From here it becomes 90 = 5αfa. Alpha Friends, 18 is here. This is at least Compared to others, it's a bit challenging was the question. Okay, let's see the angle bisector now. Friends. Definition of angle bisector we had already done it before. What was it? The ray that bisects the angle is the straight segment. Here too, of course, when you place it naturally, of course, new information future. We already know that information. What is it on? Three things in geometry If there are two, there must be a third, according to the rule We continue on our way. Friends in geometry, two angle bisectors pass through The third one will surely pass through the ground and this where the three of them pass, we use the angle bisectors the point of intersection or the inner circle we will call it center. Is it ok? Definitely most of the time you get questions like this gives two. All you have to do is It's clear that the third one also passed through here. is to see. Is it ok? From now on Look, uh, the height is two heights The third one passes where it passed. Two edges The third one passes where the middle one passes. The third angle is drawn from the point where the bisector of the two angles passes passes. Well, two side center posts The third one passes where it passed. Always pay attention to this. Or the magnificent third Look, if there are two pieces, there is also a third. We will always proceed by adhering to this. Ok Is it? So he said this once Let me be. Even if he doesn't ask a question here I will show the angle bisector. And what I said K point is the cut of the interior angle bisectors like point. Its other name is inner circle It's central, friends. Let me show you. Really inner circle from 11th grade Maybe you remember. A circle like this. TYT descends from the center straight upright. The straight line descending from the center to TYT is perpendicular. The perpendiculars to the arms of the angle bisector are equal. As you can see, such an image It's happening. The proof comes from here. The perpendiculars descending to these branches are equal to each other. This is also necessary as a companion brings an angle bisector. This is why It's simple actually but of course it's the angle bisector We will talk again. You know, a little proof? Should I leave it there? ABC 70° ADC I asked what it was. Look, my topic is this at work. From the place where two angle bisectors pass Surely the third one will pass too. If necessary you use in the question. If necessary, here You write 45. You can write 45 here too. If he gave two, I will see the third one. Because. Then write AC angle 75 70° here. It doesn't matter, I wrote 30 35 like this. Then 70 Well, since 90 is 20, 10 is added here too. I wrote. What did I want? ADC 45 10 more 55 What's left? 125° look like this We made our question. We have information here now. Memorization. Beta = alpha/ 2 + 90. When is this valid? Friends? Two interior angle bisectors when intersecting. The proof is simple actually. Theta Let's say. Let's call it theta. Come on here too x Let me say. Let me call it x here. Is it ok? Friends, how much is equal to beta + theta + x? To 180. Here in the grand triangle alpha + 2 beta + 2 things 2 theta What does + 2x equal again? That's 180 too. ABC sum of interior angles of a triangle. And here too Let me say T. Interior angles of triangle TBC total. Multiply the above by -2. In this way What's happening? -2 beta +fa = -180. From here Now I'm placing the bet on the right side at 2. I'm dividing. So what happened? 180 + alpha/ 2. So seriously, 90 + alpha/2. Here is the proof This is it. Naturally, you know, such a triangle comes from internal perspectives. Special interesting it was not an image. Let's see our question then. ABC is a triangle. When two interior angle bisectors intersect the angle formed at their intersection, What is the angle here equal to? Theta here/ Equals 2 + 90. If I call x here, x = theta/ 2 + 90. Okay? This we can say. Then I look. X what is here? 100. If I write instead 100 = theta/2 + 90. Thus 10 = theta/ 2. What came from Theta? 20. The first way is like this you can do. There is a second way Let me show you. I think the cutting of the interior angle bisectors find the angle at the point It's not actually difficult. The second way I'm doing it. Follow. Let me call it alpha here. Let me call it theta. Let me call it theta. Alpha Let me say. Is it ok? Well, I did say theta. Let me call it beta. Look now, friends, AKB due to the sum of the interior angles of the triangle in the triangle alpha + beta + 100 e= not 180 Is it? Yes. So what happened to alpha + beta? 80. Then in the larger triangle ABC triangle 2αfa + 2 beta + the angle C he asked is 180 isn't it? Yes. Well then this is 2 When I took the multiple it became 160. What about C? remained? 20°. This is the situation. The cut of the interior angle bisectors of triangle ABC point is point K. NE parallel BC, AK Equals DC. If AB is 21°, how many BAK? degree? Please try this yourself Are you? I want to say this right away. Then we will use it frequently again Our topic is me when it gives parallelism and angle bisector when you give it, look at Z automatically an isosceles triangle is obtained. Therefore note You can. I said it before I don't remember not saying that. Parallel plus the angle bisector gives me isosceles. Let me say this too. This is extra now Because I put it there, you know what I said before Let's connect it there. And once again There is no harm in doing so. This is for us It will be a great savior. Occasionally We will talk about this. Therefore, this After I said that, what does it have to do with it, sir? If you ask why you said it, then ask a question. gave you a parallel. K point interior The point of intersection of the angle bisectors is exactly here the subject I mentioned. Look at this first I want to show. It's already an angle bisector. Yes. Like this alpha alpha e Z because alpha really look DC and KD are equal. In fact, it is automatic because it is given in the question. As a result, KD also became equal to AK. In that case two inside and one outside so look 2 alpha. Again, a saving grace. From an isosceles triangle due to 2αfa. Bisector. Because A cutting the interior angle bisectors with the point If you connect the dots, you'll see 2 becomes alpha. In the question I gave ABK 21. Already the angle bisector BK. Be careful, the dot is one corner of the triangle If the intersection point of the angle bisectors meets, is an angle bisector. It would be 21. So what what shall we say now? 42 + 2αfa + 4αfa = the sum of the interior angles of a triangle is 180°. From here 6αfa = 138. What's going on with Alpha? If we divide by 6, 6 12 eee It was 18 23°ce. Well, alpha is 23. Then x which is 2αfa is also 46° has arrived. Try this yourself right away Please try. BA 100°, BFC 80 eee BDC How many degrees is 80 BFC? Now feature As you can see we already used it. Where is that? eee memorization property of angle bisector. Ok. Afterwards we saw one too Ok. But you know, every time he tells me that There is no way he can ask about memorization. HE So what did we do? A question like this we did it. Interesting. It was nice. And now such a good question. BC 100, BDC 80. Ok. Well, this is a feature like Is there a side? No. Book now Let's. Look, there are two drinks inside now. I am giving the angle bisector of the triangle. There when alpha is theta theta = 90 + alpha/ 2. What will happen? Let's think about it right now. What can I do here? Interior angles of a triangle the sum of something extra special I won't think. Look, say alpha, say alpha. What can I do that? theta d tet at least 2 If there is an unknown, work with two equations I'll handle it. I'm doing it. 100 + 2αfa + theta are the interior angles of the triangle 180° from the sum then 80 + immediately from which triangle I'm talking about right now? From BDC. How many degrees is 80 + 2 theta + alpha? Again 180°. Let's look at it from one side here. The 180s cancel each other out. Then 3αfa + 3 theta = 180°. What is alpha + theta? came? If we divide by 3, it is 60°. Alpha + theta 60° is the sum of the interior angles of the triangle BFC in the triangle There is alpha. Look, there is theta. There is also F. 180. We have already found alpha + thet. 60° What happened to F, friends? 120 degrees came. Look, it was such a good question. Let's come to another feature of ours Friends. Well, K is outside this time the point of intersection of the angle bisectors. Immediately I'm telling you. Two here outside bisector. Look, the first exterior angle bisector, second exterior angle bisector. This is called external tangent is the center of the circle. Or outside is the point of intersection of the angle bisectors. This is our specialty. Beta equals It is written here too, you know, once again Let me write it from scratch. 90 - alpha/2. This time minus sign by the way friends. From 90 We subtract alpha/2. This is actually Well, we can prove it. Let's say x, Let's say y. The sum of the interior angles of a triangle is x + y + beta 90°? Is that thing 180°? Yes. Which triangle? Here in red the triangle I painted. Then come to this. Salient angle let's do it. 180 are both exterior angles. For that reason 2x + 2y + 180 - alpha exterior angles Isn't that 360 in total? 2x outer was angle. Exterior angle at 2y. I said then Let me use 180 - alpha instead of alpha. This time I multiply the above by -2. Well, I'm checking. 2x + 2y's will go. -2 beta + 180 - alpha = e 0 happened. Ok. Then from here, uh, what would we find? Not the beta? 180 - alpha = 2 beta I divided both sides by 2. 180 - alpha/ 2 = beta. Well If I simplify it, 90 is left like this. Seriously 90 - it was alpha/2 beta. At work Friends, here are the exterior angle bisectors the sum of exterior angles by means of I used it. Interior angles in 2 red triangles I used the total. He proved the theorem We became after this. But the two exterior angles when they intersect, from the long theorem to the one that goes from interior to exterior angles half the angle at the top until it goes Just subtract it from 90. And immediately I'm telling you. Warning. Make a note of this. Look Now this information is ok but we have one more information There is more. I want you to draw this right away, too. We have a triangle, friends. Two outer We have an angle bisector. Let's do it like this. Do you know what our situation is? External 3rd from the point of intersection of the angle bisectors Let's call the angle angle A. Interior of the 3rd angle the angle passes through the middle. Look like this. AK of course Let me tell you about the triangle right away. ABC triangle. BK and CK are exterior angle bisectors. AK also pay attention and look inside the ABC triangle angle bisector. He doesn't ask questions. What are we going to do do you know? Three of these again If there are two, there is a guarantee for the third. I'm giving an example, look. First question is for you This interior angle must point to the middle not. If there are two exterior angle bisectors, the third one There is definitely. You will draw. Then look Do you know what he does sometimes? Expense drawing one. Take a drink like this angle. This time look at the interior bisector I gave it. I gave one exterior angle bisector Is it? Yes. Did they meet at this K point? Yes. Even if he doesn't ask a question, what do you say immediately? Do you know what you will say? RK here too is an exterior angle bisector. Even if he doesn't ask a question You will show it. That's the thing. With color Let me do it. I even got the red ones again I have given it to you. I gave the red ones. HE Time is here, you will do the blue too. I'm showing you another one. This is the end again exterior angle bisector of the one on the right. At work He gave that this is the interior angle bisector. Question Even if he doesn't say it, we are still outside of this blue We will show that it is an angle bisector. Ok Is it? So let's look at our question now. A, B, C, 50 etc. Those places are already I showed it. 50, 65, 40, 70 It stands. I asked what X is. Immediately you will pay attention to this here. 50 65 more what is he doing? 115. We are also good with numbers. let it be. We use it. Look, continue if you make it happen. Well, this is 65°. Bisector. 40 70 more 110. What's left? 70. This is it was also the angle bisector. Two outer angle bisector. In fact, look where I understand this from? If you think of triangle ABC, of course CD when you pay attention to an exterior angle bisector C's is happening. E exterior angle bisector of two exterior angles angle PP also passes through the angle bisector the interior angle passes through the centre. Then it was x. 50, 40, 90. So, friends. what is it? Actually, it was 45 degrees. A angle divided in half. Event This is it. That's why I definitely warn you about this. Look, it's important to note. In the question is coming to us. Now let's come here. I wrote 20 20. I wrote it point by point. X and I asked how many degrees Y is. Immediately Let's check. There is an interior angle bisector. An exterior angle bisector There is. Even if he doesn't ask a question, even if he doesn't ask a question even on CD let me make it with a different symbol CD is definitely the exterior angle bisector. Alright What do we do from here on? Now this we will say. Where two exterior angle bisectors pass It's okay now. Join X + Y to this. Alpha. What was alpha? This remainder above Half of the 3rd angle, that is, 90 - 40/2. HE What is time alpha from here? 70 degree. This is the situation. In this way Friends, we did the question. It happened in the meantime You forgot the rule. Finding X and Y you want. Let's see what you will do. Of course from where we proved the question you will try to solve. Now come to the point. I said alpha alpha. I also said theta theta. 2αfa + 2 theta + 140°, which is the outer angle of the triangle sum of angles. Come see what it is equal to? Isn't the total of these 360? In this way What happened? 2αfa + 2 theta 220 What happened to alpha + theta? 110. Now this little in the triangle alpha + theta + x + y = 180 I already know it's 110. Here you go x + y came from the interior angles of the triangle 70. I could have asked the question without any rules, but It's good to know the rule. I just watched the video While filming, the electricity went out and 2.5 Hours of video almost went to waste. Because I'm dealing with it, you know, a flow there may be a disconnection. This note I was talking about last time. Well, the connection because he went to this question obviously Was I? I already told you about 58, I know. but I need to continue from here I think so, frankly. Sloppy rather than telling. I'll do this again from the beginning Let me explain. ABC is a triangle. BD and CD bisector. BC is 2 times BDC. 2X and X We have information that: Knowledge from 0 I'm writing. Let me show you the rule first Friends. We have an image like this. 1 eee interior angle bisector of triangle ABC Let it be the green one. Is it ok? Later The exterior angle of one angle C of triangle ABC Let me connect the angle bisectors like this. This is also external bisector. An interior angle bisector, an exterior angle bisector If our angle between the angle bisectors is alpha here with alpha on the same line as the triangle the measure of the vertex angle at the same level is 2 becoming alpha. Why am I looking there? The camera was there, sorry, I was careful. dispersed. There are 2 alphas. Now then this Where does this rule come from, sir? You said it like that, but alpha 2 alpha x Let's see how 2x. Look why I'll show you but you can use this. I'm saying it again. One interior and one exterior angle the angle formed at the intersection of the median exactly half the apex angle. Alright From where? A head if you want it yourself Tired. How can we do it? So really What is the 2x relationship? The point is this Look again, don't say anything because it's a rule. You know, the teacher also sees everything as two inside and one outside. It handles it but it comes from there anyway. That's a little beauty. Well, look now here we normally don't know. This place Let's not know, okay? Let it be Y. Because, you know? We claim it is 2x. Let's see our claim is it true? I don't even use numbers. Letter I will give. I call this alpha. Here I say alpha. Now let me call this theta. like this. Let me call this theta. Look, be careful. meat please. What is x alpha two inside outside theta equal to? happened? X + alpha. Then x + alpha like this came. Now pay attention. y + 2αfa. Before I made a Pythagorean in this green triangle. Thing Where did Pythagoras come from? 2 inside and 1 outside I did. X + alpha = X + alpha theta. Now Which triangle should I do it in? That red Let's make 2 inner and one outer triangles. Come Let's see. What is y + 2αfa equal to here? is happening? At that angle. The angle I painted purple. What's that? 2 theta. 2 theta what? X + alpha's 2 times. So 2αfa + 2x. The 2 alphas are gone. Y what came? Came 2x. Have you seen? At work That's the rule. We proved it. After this Now you can do it like this. Sample now Let me show you. One such interior angle Let me make you a partner. It's the same thing Don't let it happen. Angle. One such exterior angle Let me make you a partner. I made it an external angle partner. I made it an internal angle partner. Let's say this is 40 degree? Then the angle on this line is We call it 80° like this. See this the question. Now get to work. Well, both ways we can do this question. The first is 30 I gave the angle bisector. Ok. First of all, this Let me do it from the rule I learned. A interior if there is an angle bisector, if there is an exterior angle bisector Surely the other exterior angle bisector is also from here passes. What happened here, naturally? Now he sees it immediately. I'm showing. Look here 60°. What did we say? That angle D all from D = 90 - 60/2 e became 60°. All of D is 60 is what remained? 50° left. Look, this is 50°. Now Let's see. Exterior angle bisector. Then here interior angle bisector. One inside, one outside when combined 50°. Then x from here 100. Here is what I did for you, with the rule. Is it ok? This is the first way. Just to show the rule for. Something more beautiful on the second track There is. Let me take it back like this. Just one note. Okay, now let's look at the extra mile. There is no extra distance for every question but it is equal It's really nice to deal with it when it comes to It's happening. To put it bluntly. For example Let's ponder this question. General geometric Let's comment. Now one times two where an inside and an outside pass through What did we say about time? At the same time, exterior angle BD it also passes through the middle. Ok. Then find X I wish I knew these things. Look I directed myself again. A I said can I find them then? Of course I can. 30 + 10 2 inner one 40 from outside. Look at triangle B, 30 10 Since E is already the angle bisector, It was 40° again. 40 40 80. Then what about X? remained? 100. As you can see, this is how it is We can also ask the question in this way. Now let's look at a folding technique. You may have seen it before, but 9. Because you saw triangles in class Inevitably, my friends, a little bit... there are more basic questions It's happening. And you're not ready yet but now you are ready. Folding there are really many questions for us precious. How do we solve it naturally? what about these jobs? When you fold it First of all, you will pay attention to this. One when you fold the shape in two dimensions Let me spell it out right away. Unit A side length A unit length when folded along the edge constant. Then B is a unit length side B unit length side has not changed. Where did he come from? At the beginning AB edge. Now it's a to the power of B. If we call this C, AC and CA prime are natural. became equal. Later on fold line BC is a symmetry is the axis. So when we fold it completely Y y angles overlap because they overlap does not change. Then the X angles do not change. In fact, even, oh my, the angle is 90. Because the angle won't change when I fold it Again A to the power of 90. So what should we pay attention to? Are we doing this? Because it is the axis of symmetry the angle does not change. angle when folded it becomes apparent. The second one is definitely The side lengths do not change, friends. I'm showing it here again. Look at this Let's repeat it over again. What is it? What to pay attention to when folding Are we doing this? A length does not change. Like this AB to A to the power of B equals Unit is long, B is a unit long. 2. Again, the length does not change during folding. Unit A length is A unit length is 3. Here Look, it becomes an angle bisector. The angle just now Here is the angle bisector x again as I showed don't fold it to you. Just pay attention to this of course. And the angle in folding The angle became 90° 90° again. This is our subject. The length does not change, the angle does not change either. Let's do it then. Come on now, let's see this. So, what should we look for when folding? Well While we are saying we will do it, pay attention to these: Let's. What to pay attention to when folding can we? One, there is a beautiful situation. If we do a fold like that one folded so that point A is this here the line segment AB that I colored in red came upon him. Look, I folded it. A has arrived This is how the A base became. Let me show you from 0 times naturally what will happen in BA? B settled here as his base. That's how it happened. TO then it's still really BC truth is the axis of symmetry and is actually the angle invariant bisector. Then look at the alpha alpha the angle has not changed. Did you notice? One isosceles came. Then friends when we fold the point if the line What happens if it comes at you? Bisector and isosceles angle C is definitely 90. On the other hand, you can also see it from here. In folding, due to the angle bisector, this is Tetsa and this is shouldn't it be theta? Axis of symmetry The angle does not change during folding. Bisector It's happening. So if 2 theta is 180, what is theta? happened? 90. All kinds of friends are here What will we see? On the right side too I'm showing. Now it's bigger like this Let me do it. If you fold it the point you fold onto the line If it comes here, of course the angles since it will not change 2 theta 180 theta 90. The starred part here is 90 degrees to see. Because it is already isosceles angle bisector and of course side bisector happened. And the axis of symmetry is really The length of the side of the axis of symmetry does not change. The angle does not change. Special situation here It turns out to be 90° by itself. The first one This. But always this point A AB Does it have to come to a point? No. We said it here. Look here It may also reach a point outside of this. Let's see. For example, let's say we folded it. A little overflowed out. You know, that shape from 0 to you I am producing. I bought. Length AB. Well EU I don't want to say, don't get involved. C. Here is the A prime Let me say. I folded it. AC and CA prime like this overlapped. Then I came here, for example, look I say D like this. The exponent of the language is equal. Is there an axis of symmetry in folding? Yes. Look, if I say alpha, it's alpha. Later If I say theta from the angle bisector, theta. Like this If I say beta, it's beta. Being an axis of symmetry the situation has not changed. The angles do not change I saw. Side lengths I saw that it has not changed. But the length is so a length that overflowed. Is it possible? Of course it is possible. Thirdly, the EU again it is outside but inside the triangle like this may also be in the region. Look what happened? The first time it fell on the triangle. In the second one, the folding went outside the triangle. partial. The third one is where I folded it remained in the interior of the triangle. Please, what happens? time? I am showing the same situation again. Look you will definitely and definitely pay attention First thing is the side length does not change. Secondly, the side length does not change. The third one is the angle bisector. Fourth Be careful with the angle bisector. Later The angle does not change when you fold it. Alphaeus Jesus This would be alpha. This is how it is. The side length does not change, the angle does not change. Pay attention to the axis of symmetry. Bisector comment and get the issue resolved. In this way There are different versions but especially this one part that causes such twin edges need to see. Questions are useful to us. Now look at this information In the light of this information I have given, the question We can look. Let's check it out now. Front a vertical with a yellow face and a blue back the triangle is folded along its height. 2KL = KM, how many degrees is X? I'll tell you right here. More before, when you were solving question banks I was saying this too but now the topic is explained By the way, let me tell you while it's happening. My first suggestion for folding is what you know Are you? From the original triangle, residual fold over the original quadrilateral comment on it anyway It's really nice. Therefore, I Do you know what I do? Back to its old state I'll bring it. I made the shape like this. Look Let's call this point A. Is it ok? Point A used to be here. We came K to the point. The length does not change when folding. Thus point A is on the line has arrived. Look then of course in folding the angle does not change. Theta theta 180. So 90 happened. Then again the angle bisector like this There is. Again, the angles are similar to each other equal. Be careful around the edges too T If we say TA and TK are equal. On the right and left side I also show. I folded it. Such K came to the point. When I fold it with TA TK is equal. The edge length when folded does not change. The angle when I fold it does not change. The angle when I fold it does not change. Here's the thing. What did he say now? question? 2KL KM. Then even the fraction letter I won't write. Let me say 1 unit, for example. Look Let's say it is 1 unit. Let LK be 1 unit. The edges are the same. In fact, it was twice as much as LK. KM. An image like this appeared. Meanwhile What was the initial state of our triangle? right triangle. As it is AK 2 units, KM 2 units are magnificent TK from the trio also became 2 units. Even first 2 and then 2 due to folding A situation arose that would happen. Did you notice? Equilateral triangle 2 2 this in the triangle I painted, in the TAK triangle 2 became 2. Then let's write now. From such a 60° angle bisector 30 60° all I write exaggerating. All are 90° Look at 90 - 60 for x 30°. This is the thing. Be careful when folding If you do, I hope you will like the rest, please comment. you can bring it to the house. This question immediately you're trying. Purple on the front and red on the back rectangular paper in two equal parts on the rectangular floor placed as in Figure 1. When the paper is folded along a line The image in Figure 2 is formed. To this I said how many degrees x is. Yes Let's see. Now two rectangle tops There is a blue rectangle on top. There is a blue vertical thing, a purple rectangle, sorry. Part of the purple rectangle folding. The back side is also red. Now what happened naturally? The first one Look. Be careful. Back to its old state Even if I brought it, the image is already like this like this. It was here in its old form. New red part in case. What will you say? Beware of. If it is 90 degrees, the angle does not change. 90° angle bisector. Beware of. Bisector. Again, such angles are equal. Now what? So what will we do? Just think about it. Two rectangles on top of each other. What is the property of a rectangle? Opposite the sides are parallel to each other. Opposite the sides are equal and the angles are 90 degrees at the corners degree. Is this information useful to me? What can I do it? What does it mean to find X? What do I actually need to find X? there is? On the other hand, how to get 100 will I use? I'm telling you right now. Thanks to parallelism we realize that you should have done it first. X Z because X. O What happened to time friends? My X to find this red that I actually painted I need the angle. Look, this is red. to the angle. Because it was 100 degrees at that time since the corner is a rectangle, it is 90°. Like this If I say a theta here on the edge too Let me show you. Let me give you the letters. Let's call it F. F R P. What happened? I painted 90° red angle and point P and on the other hand o How many degrees is the angle P at the point I painted? 100°. Then the two inside from outside to here theta When I said "theta + 90" what came up? 100. So what is theta? 10° theta is now 10° I attached it here. Subtract 10 from 180 and get 75 Divide by 2 from the angle bisector 85. Due to Z What exactly was X anyway? This is it. Well Our answer was 85°. This is how it is We have solved this question nicely. Yes Let's look at our question. The preliminary given in Figure 1 ABC with yellow on the face and blue on the back triangle shaped cardboard AD and FE shape when folded along the lines The view shown in 2 is obtained. In Figure 2 50 degrees formed in paper-folded paper shown with X according to the angle measure What is the measure of the angle in degrees? Yes. One Details will save us here anyway. Now let's say we're going to make a fold. Is it ok? Point A is the same line What happens if the A on it comes on top of it? The length does not change when folding. In folding the angle does not change. In fact, without you even noticing what gives here? It gives 90 degrees. Please be careful with him. Because normally when you fold something, for example I give you, we have an edge like this. Ok Is it? That edge. You put this somewhere Let's say alpha when folded like this let it be. You took it and folded it somewhere. Normally It may look like this but it looks like this in this way a linear point A if it comes on you where you folded it In fact, there is a symmetry. Two There is a skener triangle. Be careful with that too. From now on, always such details when it comes to the same line here is 90 and here is see that it is an isosceles triangle is waiting for you. That's why here too Notice where point B came from? To E. HE What will you see when? Actually, really As I just said, BDE is linear. for 90° will be 90°. BD equals DE future. The length does not change when folding. EU because it sits on top of AE become equal to each other. Then the same again Look at the figure, where is point C? Same to point A on the line. There At the folded point the angles are equal. Again What message does it give you? For that reason He wants you to see it be 90° 90°. Then again the angle does not change. Alpha here If you say, look, this is alpha. I'm throwing it here if you say theta, theta from the isosceles triangle. Pay attention to these. What else can we say? Now let's get to the question. Here it is, back to its old state When you bring me here, I will be I saw that the angle was 90°. Old When you make it here, I am now I called myself alpha. Please be careful. There's a right triangle here. Have you seen? When you get a right triangle, 90 What's left for 50 and alpha? 40° left. This So, what is the benefit to me, sir? This is it. When I fold EC over EA doesn't sit? Then isosceles triangle and doesn't the angle here also become 40? Iqsh What else is coming from the outside, 40 40 X? 80. Look at this beautiful thing detail. AB = AC. BE = EC. given. ABC triangle paper in the form of arrow along DE when folded Corner C is the height of side DC coincides with point F on it. ADF What do we do since it is 34°? What is X? We say it is a degree. Yes. Please try Let's look at this question right away. Actually Isn't our interpretation clear enough? Isosceles in a triangle If it is equal to ABC, tap excuse me, a median coming down from the top is the same is the height and angle bisector at time. So this is the first one. This is the second one Going back to the question, what is point F? It was a point on the height. TO Friends, then what must A be? Linear because let's say linear Let's assume it's not. E is isosceles in a triangle the perpendicular goes to the base and the same time must be divided into two. to E 2 divisor correct FE. And this is also above the height then AFE linear. Here's the thing. Realizing this important. After all, this place was 90° then. And look, it became 90°. More importantly The edge does not change during folding. Ok. The edge does not change during folding. E and ef equal to each other. Ok. But also the angle bisector is the axis of symmetry in folding of course ED. What happens then? At angle D angle bisector, angle bisector is formed at angle E and thus 45° becomes 45°. Don't neglect this place either. Let's not. Where was the ADF? This was the ADF. Look, we showed it as 34°. Where I want? I want X. After this Since I see the ADF at 34 degrees, it is again use the angle bisector. Here if you need it. HE What is he doing? Well, let's see right now. 34 out of 180. 146 73° 73° our subject This. After that eee 73 45 more 118 What's left here? 62. 62 thick 62 90 Our angle at the top is 208° It happened, didn't it? Yes 28. Of course angle bisector height is also This is also 28. So our correct answer is 56 happened. Yes, that was a good question too. After all, he help with height relationship etc. The staff always ask questions from time to time I will try to pay attention. And also in folding, the axis of symmetry is the angle bisector don't neglect it. Yes, let's see. Now the same colored sticks are of equal length two each of green and pink, one with yellow and blue colored sticks The resulting figure is given below. Linear green bars yellow bar green at the point where it meets 15 and 15 perpendicular to the bars Two 45 degree angles are obtained. To this What is the angle represented by X in degrees? Yes, let's look at it together. Once upon a time greens are equal like this. Look there two important green bars. He sees Are you? Here then these two green Of course, if there are 190 degrees on the bar What would we say without any doubt? Immediately we will see the isosceles there. Like this If I combine Let me spell it out right away. AB and AC are equal. BC is also equal. Because AC and BC are the questions themselves gave. I also discovered a new isosceles triangle. This is how I saw the EU. If it is 45°ce here 45°. Yes, it is such a nice situation. Now An important move in the question, friends. Now look at the angle between AB and BC. degree output? 45 + 15 to 60. So what? will we see there? I immediately AC I will combine. Because there are two edges If the angle between them is 60 degrees, there is a hidden there is an equilateral. In this way, AC is also tab AC I shouldn't say AD. AD also called them became equal. Equilateral triangle AD AB BD created and we need to write it here immediately if 60° What do we say afterward? One Look at one more isosceles triangle here output. Do you see him? AD and AC. Have you seen? Then let's continue. Now Normally it was 45 degrees. From where? 90°. 45 45. Then because of the equilateral triangle if necessary, click on the red one there I'm writing down the tiny angle I painted right away 15°. What else should you pay attention to? 45°. The 45° top angle is already 90°. Then never Even without using 45, this we can say. Because of the equilateral This is what I painted pink, this is what I painted purple the entire angle is 60. Since the entire angle A is 90°, 30° remained. Now AD is red due to equilateral the edge came out. AC is already on the red edge isosceles triangle appeared. Our top angle is 30° Since it is isosceles 75. Like this Let me show you by painting. This is our angle too 75. Here the question is over. 75 60 more x 135 It came like this. Look at that rhombus in between We used it and a good situation occurred. 135 happened. The isosceles triangle given in Figure 1 3 of the plates in the form of into a triangular region as in 2 with one edge overlapping when placed x degrees equal angles are obtained. According to this, what is x? degree? Yes. We have an extra situation is there? No. In essence, only in forms What will we pay attention to? The angles are the same that it is. So this x is also top angle x is also x here. What should I pay attention to next? Of course, because it is an isosceles triangle Let me write a y here. Take a look. What is equal to 3x x plus 4x 4x + y? 180°ce. Sum of interior angles of a triangle. Meanwhile He also secretly called the base angle y I became. Isosceles triangle smaller isosceles the base angle of the triangle is then 2 vertex angles x. Of course, this is also 180°. Then here it is Let's multiply by -2 to cancel out the y's for. -7x = -180. What happened as you see in X? 180/7 came. Guys, this question is my fault. It completely escaped my notice. New when transferring the print the letters here There was information here that we would delete it. Him They forgot to write it down, but I can't blame them. After all, they are doing something written. Me too I'm checking. ultimately knowledge I didn't check. Where is the shape here? That information was overlooked because we changed it. Please write 20 degrees here and answer the question Can you solve it? Solve accordingly. Ok Is it? As I said, that information is completely there was in the past. In the new edition, it is in the form of letters there were letters here too. A B C D E saying. No letters. Where is the ÖSYM format? While I was trying to get it, this time I added the following information below We forgot. I'm sorry. One solves Are you according to this? Just because There is a deficiency. The square given in Figure 1 4 identical except for their colors inside right triangle corners with the same angle measurement overlap and space between them placed in a way that does not remain The view in Figure 2 is obtained. Accordingly, shown with x in figure 2 What is the measure of the angle in degrees? Yes. Now You need to pay attention to this here. A steep The longest side in a triangle is the hypotenuse. HE Let me show you this time. I have to show this in red hypotenuse. Because look, pay attention. These Isn't it the same shape? Four identical triangle. As you can see the hypotenuses are like this same. Then the hypotenuse, the hypotenuse, hypotenuse, hypotenuse. Because outside It overflowed a little. Then of course here angle 90. The angle here is 90, the angle here is 90. Of course, there is no deficiency there. Him you would see. Therefore 90 90 let me tell you that. Because this way, we should also see. After all, look, this is the hypotenuse. At work this is the hypotenuse. This is the hypotenuse. Where are we going, sir? From here to here hypotenuse. Look now even this is something should tell you but 20° 20° and 20°. Let me enlarge it now, see. Hypotenuses because it is equal to 20 20. Here What did the angle I colored in red turn out to be? 60° output. by equality of hypotenuses here is an isosceles or even an isosceles If the vertex angle of the triangle is 60 degrees, What's hidden there? Now say that too Let's complete it. Of course there is a There is an equilateral triangle. Let me delete these see first. Equilateral triangle. Look 60 60 This is 60. Now let's go back. Here because I'm going to show you one more thing. These did their job, I'm deleting them. This is also It did its job, I'm deleting it. Even this job I did it and deleted it. I'm deleting this place too Friends. Here we have an isosceles triangle. Did you notice? The red edges are equal. So what if I combine it like this? future? 20. Let's do it right away. 20. This is 80 and this is 80. No need for this Even though it doesn't even remain, I can delete it. Empty give. No need when you're a twin. This is already here 60° due to equilateral. Naturally of course x to be 90°, 20°, 70° 10° left. This is the situation. In this way x Look, we found 10. I was going to say isosceles but I gave up because the red triangle isosceles but if he had asked something else If necessary, these two edges here the isosceles interpretation of the equation I would do it. But as you can see, this is the case We've got it sorted. The right triangle in Figure 1 is magenta the part of the right triangle that is colored the same like figure 2 when placed two triangles are doubled the edge overlaps. The 15 degree angle given in Figure 2 What is the angle shown according to the measurement? degree? Now here we just actually follow the shape. It is enough to do. Let me show you with letters. Look where I moved the AB edge? Here To the edge of the EU. Then be careful in the beginning This AB edge was already standing. In that case a hidden isosceles appeared here. Oh Let's pay attention to it. Is it ok? Initially AB edge. Look, it really does. Myself I wrote. Then when he came here came here. This is how the EU border turned out. Is there another kind of equality? No. This at least it's red. Red is there. Let me show you the green one too. In its old state de green was here. There is no connection between them at the moment but now it's a move like this we can do it. Since AB is isosceles, AB saying. Let me call it alpha here. Afterwards Since it is an isosceles triangle, alpha + 15 Let's say. From now on, also the angles But let's be careful, okay? These We wrote it down according to the triangle I'm carrying now. Let's not neglect it. Here is such a theta Let me say the angle. Before the green Since it is opposite to angle B, then angle B is also will be theta. Look, be careful, it's very important. I'm saying it again. Now this I gave theta angle a letter myself. Is it ok? First, now theta in the magenta triangle against what? Of green. In that case What is this opposed to in the beginning? Theta. Across the green. Why this specifically? I emphasized? Now please pay attention. What did I just call this part here, angle B? did I write? I wrote alpha. Now what? did I write? Theta. E alpha tet equals. In that case I can also write back theta here. That was the situation. Therefore, here Theta of course the reds here Theta is the angle between. Then alpha I wrote theta here instead. Theta + 15. The sum of the interior angles of a triangle is 3 theta + What did theta come from 30 = 180? 50°. Me too What do I want? I want X. Tet 50 From what I found, there are already 90. What about X? remained? 40 degrees left. This is how I improve. I might even ask a question. As you can see below is the red line a fixed point A with parts and positions with three different shapes of a fixed square image is given. Yes. An endeavor Let me tell you accordingly. Look at the fixed point A and the position of the square is fixed. So here it is What did we say about the red one? Three different There is an image. The red line segment is also There are images. In fact, completely something we will do with the isosceles interpretation. If it is equal to one side of the square, it is red What will happen to the edge here? Of course isosceles triangle. 25° 90 eee 100 30 What remains to be done? It remains 40°. What should we say next, sir? Then A the point is fixed or actually here If I say point B, it is BA anyway. equal to one side of the square. 40 over there because of the degree this is an isosceles triangle. 70 70 Point A was fixed, look it is not changing. That's why it became isosceles. In this way 70 90 What's left in X? 20° left. Like this In fact, if we are careful, we can do was a question. Two identical isosceles triangles. Someone's one side BC of equilateral triangle ABC one side AB on the other parallel to the edge is being placed. Accordingly, I asked how many degrees X is. Yes. Let's deal with this too. Now an equilateral triangle measures 60 degrees like this Let me write. Equilateral triangle 60° Let me write. Edges It is not necessary, at least in this question. Like this Let me do it. Well, but I'll put it there anyway. Such edges are identical to each other. What what will we do next? Come on, even Let me show you this too. Identical iskender triangle. I did these. We showed it like this. What did he say was the problem? To me? Be careful. Parallel to side AB. HE Look, I'm showing time. To the edge of the EU parallel. Which one then? Of course this isn't it? There is no way you can go and do this. HE Since these are parallel, how Can I make a comment? AB and black two edges parallel to each other. Then let's think about it right now. I say this parallel or corresponding angles are the same 60°. Okay, I wrote it like this. Companion because the angles are the same. Then please pay attention an angle of an isosceles triangle is 60 + x base angle. Then the identical isosceles Since it's a triangle, what's this angle? 60 It became +x. Is it ok? That's what we did. Later corresponding angles are equal to each other. 60 again It became +x. Now please pay attention. Just one Let me phosphoresce. To that green again Please pay attention to where I highlight Let me see. See that 60 + x here? 2 inside and one outside will work for me. I'm writing this once. Come on a long time ago Let me call the top angle Y. This is what I painted in a triangle y + 60 two inside and one outside 60 + Isn't that equal to x? Let me show you like this Look. What is 60 + y equal to? Of course here the 3rd non-adjacent exterior angle is 60 + x, that is, if the 60s go, what does x come to? It equals Y. I'm deleting. Only The unknown is now actually x degrees here. Is that okay, teacher? This is how You showed me that you are definitely You showed it with intention. This a little more Can't we show it in a different way? Yes, we will. Look, always with this intention I won't show it. Let me tell you that. Each There is no such simple way to solve the problem. Every question It can't be like that. But in this question the second way the second way to find this place I'm telling you. 60 + x 60 more 120 + x. What remained? That place I painted red? 60 -x. The sum of the interior angles of a triangle is 180°. This information. By the way, the triangle is an equilateral triangle Since ABC is 60 in total. 60 out of 60 - If I take out the X, what's left? This X to the angle I painted green. As you can see any kind of green angle here x degrees We found it. Then the base of the isosceles triangle is angle was 60 + x. We just told you. Base angle 60 + x, top angle x. This thus 120 + 3x = 180°. 20° from X 60/3 happened. This is the answer to our beautiful question. Now these are the last two pages now page thing these two questions on the last page one Let me see. To the two questions on this last page When you look at it you can say something. Oh Since the teacher put the most difficult questions at the end He put it. No, I don't have that style. Mixed when I move on to the questions, that is, from 65 in other tests such as the next I will do it. The topic is over. Now, you know, like this When I said let's solve 78 extra questions their level is karma. Is it ok? Because Isn't that the case in the exam too? Question 32 difficult, question 36 is easy. Sometimes here are 40 questions It's very easy but it gets harder with 32 to 33 questions. It's happening. That's why the level in karma We have no problems. Difficult easy medium in between. Now that I have had so much conversation So this question is not that difficult. Read it. Eyüp took it on holiday three identical isosceles triangles Place the magnet on the refrigerator base corners and one edge will overlap when you paste it as shown in the figure 144 between the edges of the magnets degree angle is formed. In the last case one of the rhombic sides of the yellow magnet parallel to the bottom edge of the refrigerator door Since the green magnet is the same one of the edges of the bottom of the cover What is the angle x formed by the side degree? Yes, again, ÖSYM style text There is. We will start from the text. Oh my text insignificance. The answer is hidden in the text. Because look what he said? From the equilateral edges one yellow magnet on the bottom edge of the cover parallel. Well, of course, this crooked edge Since it will not be parallel, which one will be parallel? This is parallelism. What is here? Is it useful to me? Look at the EU edge parallel to the bottom edge. CD like this Let me say. Parallel to CD. What happens? In the right Yes, if it is 144 due to angles. Something like this As you can see from 180 - 144 rule C is 100 44 36 came. Then find my x the angle for which I made this question mark I need to find. So can we find it? How? If you haven't tried it yet, please try. I have a chance. To find my x this I need to find the red angle. How? These are identical isosceles triangles. This is called y If I say, look, this is y. E naturally of course congruent triangle Y. Congruent triangle Y. Notice? What's here again? there is? 360°. 3y + 144 = 360. From here 216/3 Y 72. When Y is 72, the whole straight angle is 72. 36 more 108. What's left for X? 72. This is it in this way. Actually it's an easy question but what I care about is 360 degree parallelism and We used the identity very well. We're finishing with the last simple question. Side view shown in the figure the same as a rectangular glass sized window on the top edge can be opened and closed around. Intermittent when the window is opened x degrees the line segment formed by the horizontal angle y is degrees. Accordingly, with x The relationship between y is given by the following value which one is it? I mean, here, of course. Let's not say which of the following because it is not a test. Well, that's not how I wrote it here's why. Friends in the test exam of course, which of the following future? And this is in this ÖSYM format asks. Which of the following is it? Let's say and create it. Look I didn't even give the options so that nothing would happen. This is for sure, sir, no need to talk about it. Let's do it properly. From each other's kind Not all questions are difficult. Carefully to be and to do according to the rule is sufficient. What do we say now? Once I rotate the length constant. Come to the simplicity of the question. Look Now I did it like this. What did I say next? Don't be afraid of the isosceles triangle. Sometimes here write it down 90 - x/2 write down 90 - x/ 2. Mis like. Is it ok? Well now it's time to hit the wall steep. Naturally 90° 90 - x/2 + y = 90. What is the sum of y from here? x/2. Well What is 2y? It was x. Or x = 2y It could also be y = x/2. X and y the relationship between the options. This too It's true if you see it, it's true if you see this too. This is the first way. The second way is this, you say. Sir, this place is already 90°. 90 - y I wrote it here. I wrote 90 - y here. This The sum of the interior angles of a triangle is 90 - y 90 - y more 180 - 2y + x = 180's gone x came to 2y again. Here is the second way again Just what do we pay attention to? In rotation Oh, the length does not change. Be careful with him. Is it ok? Let me show you this again. Look the length never changes. New isosceles triangles appear. And a letter We will not be afraid of questions. Neatly every We will continue to do what we do. At work The explanation part of our topic is here I'm finished. What do you do now? Classically, definitely your question bank You are passing. Look at the angles in a triangle. We reserved enough space. Because it is very valuable here. We are building. loop by loop we are weaving. So of course the angle in the triangle should have lasted longer than most topics. Now we have made the angles. Two on the subject. Where is the time now? Steep in the triangle. See you at the right triangle Friends. Enjoy your work. Yeah.