Transcript for:
Angles in a Circle

uh okay good afternoon so let us start today's lesson it is a very simple but very important lesson uh so we will finish this lesson today angles in the circle so you already know what a circle is there are some terminologies of course I don't know the Swedish words I want you to learn them yourself if you want to but the English words I can mention it here for you so if I have a circle then you have the center for example let me call it a c or o for example and then you know that there are uh radius to a circle yes so for example let me denote it by R and you know that it doesn't matter which point on the circle I choose if I connect it to the center the length is the same as the previous one for example if I take this point and connected here if I have a circle indeed then this length is also equal to the same here all of them are R okay so that's the radius what is the diameter yeah so the diameter is a line that passes through the center and cuts the uh Circle at two points so this blue line segment that you see is an example of a diameter if I ask you what the length of the diameter is and of course it is clear it is two times the radius okay another terminology that I want you to know is that if I have one piece of a circle separated this one is called an arc yes so a piece of a circle is called an arc the biggest Arc is what the biggest AR the biggest Arc is the circle itself yes so that's the biggest possible Arc but you can have smaller arcs that is something that I want you to know and then if you connect this one if you have a if you have an arc and then you take these points and connect it to the center you get one piece of your circle that piece is called a sector okay so I just want you to know what is what are these words so if you have something like a piece of an apple pie or piece of pizza that is called a sect and there is another terminology that I wanted to know and that is Arc Length so what do we mean by Arc Length so let us say that this is an arc and if I want to denote it usually I denote it by a but then I put a something curved on top to show that I am talking about an arc not a line segment because if I have two points on the circle I can connect them uh along the circle itself it becomes an arc or I can connect them as line segments through within the circle that is not an arc anymore but there is also a name for it this blue one is now called a chord okay so and then you see that the cord corresponding to these two points seems to be shorter than the arc yes and then what do we mean by Arc Length it means if you want to imagine what do we mean by Arc Length it means that you can imagine an ant moving on the circle the distance that that insect ant is moving on this Arc is called The Arc Length yes okay so what is Arc Length I told you that for example example if an an you can assume an ant is moving on the circle and then that ant moves from here to here the distance traveled by that end is called the arc length in other words you can imagine it in a different way so let us have a piece of a string I put one uh uh tip of the string here and then I bend the string in order in a way that it completely overlaps with this Arc and when I reach here I cut the string then I will have a curly I mean I will have a Bend or curve the string then I make it taut stretched when it is stretched I measure it with the ruler the length that I get is called The Arc Length yes is that clear so the Arc Length you can imagine it as the distance traveled by that end or you can imagine that you have a piece of thread and then you put it here and then stretch it it and then measure the length that length is called The Arc Length so these are very simple words I want you to learn so Arc sector Arc Length cord and center radius and diameters this is more or less the same thing that you already knew but the interesting thing that we want to learn today is about angles in a circle there are two types of angles we discuss about about them within a circle one angle is called a central angle central angles I have to tell you what they mean we also have another type of angles that lesson today is about those two angles inscribed angles yes okay so let me tell you what is a central angle let me tell you what is an inscribed angle they are very simple concept so if I have a if I have a Circle every Circle has a center so let me call this C if I connect this Center to two points of my circle this one and that one it makes an angle that angle is called a central angle so if someone asks you what is the meaning of a central angle you say that it is an angle whose vertex is the center and its arms are the radi of the circle is that right so this Alpha if I call it it's a central angle and if this one is a and this one is B we can say that this Arc is subtended by the central angle Alpha yes or Alpha subtend of Arc AB this the terminology you can say I mean this is an English word for saying just an arc opposite to Alpha if you don't like this word that's exactly the same meaning so every central angle is an a central angle is an angle whose vertex is the center and the arms of the angle are the radi of that Circle so in that regard Alpha is a central angle but there is another central angle with the same Center and with the same arms what is the other central angle this angle yes this angle which I call it beta that is also a central angle with the same vertex with the same arms but look it from the other direction so I have two central angles here one of them is Alpha one of them is beta but the difference is that Alpha subtains which AR the smaller blue arc but beta subtains the bigger black one yes I can say the bigger black Arc is opposite to Beta this is smaller blue arc is opposite to Alpha but they usually in English they don't use the word opposite even though that's completely acceptable and understandable they use the word subtended okay that's it this is the central angle any questions for example do you think that if I draw a diameter I can have a central angle what let us try understand it so if I have a center and I draw a a diameter any random diameter that you like okay do you see that I have a central angle actually I have two central angles here again what are they what are the sizes of those central angles before me telling you 180 yes because this is one central angle is this is central angle yes because the vertex is the center and the arms are radi of the circle so that is a central angle but this is one particular central angle that you already know the size of it what is the size 180° and what is the other one it's also 180° in this case so can you tell me what is the uh biggest size for a central angle 3 360° the whole circle yes so these are very simple ideas but what is the meaning of an inscribed angle that's also very simple idea so a central an inscribed angle when I want to talk about an angle I need to tell you where the vertex is and what the arms are the vertex is no longer the center the vertex is somewhere on the circle so I will come to your question so for example here I choose this one and the arms are not radi the arms are cords okay so what do I mean by that for example this one this blue line I gave it name to you what was the name the name was a chord and then I can draw another chord yes so you see the vertex is on the circle and the arms of the an angle are not radi they are quarts and then they will give me an angle between them that angle Alpha is called an inscribed angle yes and then if I ask you what can I say about this Arc again you say this you use the same word you say that the arc AB is the arc subtended by Alpha or Alpha subtains this Arc that's the notion of an inscribed angle so the difference you want to repeat in a in a central angle the vertex is always what it's a center and the arms are radi but when we talk about an inscribed angle then what should we say we say that the vertex should be somewhere on the circle and then the uh arms are cord here if I ask you how many central angles you can imagine infinitely many how many inscribed angles you can imagine how many infinitely many for example the other one can be this one this is another central angle so let me call it data sorry an inscrib angle I can choose this one and if you want I can connect connected to this one I can for example call it GMA all these angles that you see they are inscribed yes questions you had questions yeah okay I told you sorry yes subtended is a word it means opposite here if you don't like the word it means that if I ask you which Arc is opposite to Alpha you will show me this Arc we say that this Arc is subtended by alpha or Alpha subtends this Arc if I ask you which Arc subtains gamma it means that which Arc is opposite to gamma again again same AB but here of course which Arc is subtended by Beta C for example that that's just the name it's not important but what is important is this very interesting property that we want to discover today so even if I don't know what Alpha I'm gamma are I am sure that they are equal in size okay so we want to consider these things that's a theorem we want to prove that if I have any two uh inscribed angle opposite to the same Arc the sizes are the same so if for example this this is 30° this is definitely 30° it is not trivial we want to prove this okay and another interesting property is a relation between a central angle and an inscribed angle okay that there's also a very nice relation between them so we want to prove these two for example if I give you a central angle any angle that you like for example let me call it Alpha and then I draw a an inscribed angle but not any random inscribed angle an inscribed angle this vertex can be anywhere but I demand that it subtains the same Arc as the central angle subtains what do I mean by that I mean for example here and here okay if this angle is Alpha let me call this angle beta I want you to understand the difference this angle is Central this angle is inscribed but what they have in common they have the same Arc opposite to them if this happens the size of the central angle and the size of the inscribed angle are very nicely related this one is exactly two times larger than this one okay uh we don't know why yet I just want to tell you the results but then we go through understanding why why this is the case Okay so this is something I want to show what do I want to show I want to show that if I have two angles one Central one inscrib but both of them subtain the same Arc the size of the central one is exactly not approximately it's exactly two times larger than the size of the in one this is something that we want to show yes always in the middle C middle means Central Center I don't know what in Swedish they said mid point but in English we say Center yes so it is yes this one is always Center and the arms are radi but here the vertex is somewhere on the circle yes and the arms are cords not R yes R what can radi no no no no radius cannot be cord because cord has two sides on the circle has sorry has two points on the circle the radius has one point on the circle one point on the center yes that's not correct okay let us try to understand this yes no we haven't proven that yet we want to prove it I just gave you the message okay and then I want you to understand what is the point we want to understand why this is the case yes yeah I know yes but I think yeah we can change okay so back to the lesson let us consider some cases that I want to convince you let us try to do this uh let me uh take this to be Center and draw a diameter and then draw this line here okay and then uh let me also draw this line okay so let us try to test your understanding this is the center and let me call this a and b this angle I want to call it angle X this angle I want to call it angle y if I ask you what is the name of the angle X let us just review what we learned what is the name of angle X it's a central angle why because the vertex is the center and the arms are radi and then if what type of angle is y inscribed why because the vertex is on the circle and this arm is a cord this arm is a diameter but every diameter is a cord yes not every radius is a cord this diameter is a cord so that is a inscribed angle we want to show that it is very simple I want to show that the size of x is two times larger than the size of Y this is very simple of course this is one special case that is a little bit harder case I will come back to it but that is one special case can you tell me why this is correct it is not hard to understand why this is correct you can come up with the answer yourself why this x that you see here is two times larger than this Y no why it is exactly two times you might feel that yes this is bigger okay but yes you can say that because this is closer here it is wider so it is larger I agree with this intuition a little bit y I meadi away from mm so this is R you I agree what is this length this is R there is another length which is also R which one is that c c to a is also R okay so now I ask your opinion about this triangle I might you might have forgotten this English word you can say the Swedish word what is the what is the name of this triangle you see this leg and that are exactly the same size if you have what yeah it is isoceles in English it's an isoceles triangle yes okay if this is an isoceles triangle let me ask your opinion about the size of this angle what is the size of this angle same as same as y exactly because this is you know if you have a isoc isoceles triangle if this if this size is equal to that size if I call this Alpha I am allowed to call this one alpha as well yes if I have a triangle with two sides equal the angles up close to those sides are also equal so now if this triangle is an isoceles triangle which it is because this is R this is R so if this angle you say it is y I am allowed to call this angle as well y yes yes or no okay but now let me call this angle Z which I don't know but you remember MATC mat1 C we learned something about the sum of the angles in the triangles what is that so y plus Z Plus another Y is supposed to be 180 this we this we learned and then y + y is what 2 Y and then plus Z what is z z has some relation with X what is that relation Z Plus X yes so it means that if I want so Z Plus X is 180 so if I ask you to write something something for Z you move x to the other side so Z becomes what 180 minus X so here y + Y 2 y but instead of Z I am allowed to write 180 minus X yes so you see y + y is 2 y instead of Z as you told me I wrote 180- X and then after that I I have equality and after equality I have 180 now you can see what is happening I move 180 and minus x to the other side so I leave 2 y here I have 180 there I move this 180 to the other side it becomes 180 I move minus x to the other side it becomes plus X and then what happens this one and that one are gone so 2 y becomes equal to so this is always the case that the size of this is two times larger than that one but it does not cover the whole scenario this is not exactly the picture that I drew there for you yes but we want to say that no even for that case this rule is true okay so let's try to understand is that any question here okay but now let us try to make it more interesting uh so let us try to uh formulate the problem in this way I draw another picture so let us say that I have this circle here uh let me draw one one central angle here and then let me draw one uh inscribed angle but subtending the same Arc so for example let me choose the point here yes so let me call this one as before y let me call This One X okay and this is the center let us review things again and if I ask you what type of angle is X what do you say it is Central what type of angle is y squ we want to prove the same thing again because if I ask you what is is the arc subtended by X you will show me this black curve if I ask you what is the arc subtended by y you show me the same black curve so it means that it is an inscribed angle this is a central angle but both of them are subtending the same Arc this is important not just random ones we want to prove that again the size of this one is still two times larger than the size of that one okay so how we can convince ourselves for this one it is not as trivial as the previous one but still it is true we want to explore that yes are we allow to draw yes yes we you are allowed to draw anything yes yes what do you want me to do I can name them so that you can talk about the names to me okay a b c for example now C is reserved here so if you don't mind let me call it D okay Court okay so what do you think I can do here yes Ahmed is right we have to draw a line somehow from A to B from A to B is it helpful let us check I don't know so you are telling me that connect these two okay if I connect them is there any way that you can solve the problem okay so Ahmed you don't defend yourself no I'm just trying okay okay so what do you say from C don't say why D what yes if I if I draw this one to that one okay what happens this which one becomes isos name it yes this one and that one but be careful they are not of the same sizes yes so you cannot say that when I draw that this part is y over two that part is y over two there is no guarant for that if this if you have this angle if this line happens to be exactly in the middle yes I agree this becomes y/2 this becomes y over2 but there is no guarantee might be that line is passing a little bit to the left then this part is is smaller than y/2 this part is bigger than y/2 or it might happen that it passes a little bit to the right then this angle becomes bigger than y/2 that angle so you cannot say that this is y/2 this is y/2 yes but you know that the sums are what yes I I agree with you this is a good thing to do because instead of Y if you don't mind let me write Alpha and let me write here beta one thing I know I know I don't know Alpha I don't know beta but I know what is Alpha plus beta what is that it is the old y that we had so remember that I don't want to draw I don't want to write why to not mess it up but you understand that that was named Alpha I removed it in my head I had two parts Alpha and beta I don't know if Alpha and beta are equal but I know something the sum is one okay what should I do so then what if that is Alpha can you tell me yourself what is this now also Alpha do you agree because now this triangle that you see here is an isoc triangle why because the length from here to to here is one radius the length from here to here is also one radius so these two lengths are equal so if I call this Alpha I am allowed to call this one alpha to yes with the same idea if I call this beta am I allowed to call this beta as well yes because for the same reason this length is one radius this length is also one radius so the right side triangle is also I saw this triangle yeah it is very good then what you are very close I would say if I were you I wouldn't stop my line there I preferred to continue this uh red line that you told me to draw until it goes here let me call this e yes then this x is also split into two parts not necessarily equal parts but it's also split into two parts so let me clean it and let me write for example here Theta is another Greek letter and row I don't know what Theta is I don't know what what row is but I know what theta plus row is X now everything is very simple I hope you understand that yes can you tell me what is this data according to this picture can you compare can you compare the left hand side of this picture with the picture you have here yes yes why here Alpha is playing the role of this Y and Theta is playing the role of this x but in this picture I prove that X is 2 times larger than y so then what does it mean here can you tell me it means Theta which is the counterpart of X in this picture is two times larger than what Alpha yes but then don't look at the left picture look at the right picture the right picture is the same picture yes but reflected here but the rules are the same so what can I write for row what is row row is 2 Beta yes and now start adding them side by side it becomes theta plus row is equal to 2 Alpha + 2 Beta but 2 Alpha + 2 Beta I can Factor two out it becomes Alpha plus beta yes so what is theta plus Ru it is x equal to 2 what is Alpha plus beta it's y so still it is correct so you see the first one is important you start with this picture to convince yourself that if one line is diameter this is two times larger than that one and then as Omar mentioned it was a good trick to do the same thing here by drawing this diameter if you are a little bit smart to draw this diameter then you understand that you have two versions of this problem you don't need to start everything from scratch you can use the result here and then convince is that right but of course that is also not the whole story might be the mysterious one is still left there it's a little bit harder okay so let us go back here and convince ourselves that this is also the case and then we are done for all possible cases why it is done because here we considered the black uh the the central uh the central angle within the inscribed angle and then here we considered one uh common arm for both of them and here we are considering that the inscribed Circle angle is completely outside the central angle but we still want to prove the same thing it's a little bit harder to understand what to do here but then it is interesting to see that this is also working for example can you convince me that this is indeed the case we want to prove the same thing I want to prove that in this picture again the size of Alpha is 2 * s beta but I want to repeat again everything what is the name of alpha what type of angle is that Central why because the vertex is the center and the arms are radi what type of angle is beta is inscribed why because the vertex is on the circle and the arms are cords I want to show that again and then it is important the most important part I missed sorry what is the most important important part these two should subtain the same Arc not different arcs yes if I have different arcs for example here I cannot say that Alpha is 2 * beta or whatever no because sorry I cannot for example draw the central angle here I cannot say that this angle is two times this one this is correct because they are subtain in the same Arc but I cannot say that this one is two times larger than that one because this one is subtending this Arc this one is subtending the other another Arc it is important that they subtend the same Arc in that case the size of the central one no matter how you draw it is two times larger than the size of the inscribed one okay if we can do this one as well then we are convinced that in all cases this is working but how can I convince myself in this case as well you you have to draw a line again but which line do you think is useful here let me give the name for you so that if you want to name them you can call them a b c is already reserved here D yes do you have any opinion is that yes okay so let us see is there any okay no worries even I don't know we draw and see what happens so you are saying that I draw a line from here to here and then continue okay so if you don't mind let me make it dotted it means that I am adding something to my picture Okay okay so do you think it is helpful yeah think a little bit this one okay I did the I Nam after you the diamer the I Nam that one z z uh the little next to the X that's the X where is X yeah X in the middle ah so your X is in the middle yes no no no down like the angle the angle X it's Alpha okay left of it I named the angle created angle yes this one no left angle yes yes exactly that one Z and then on the right that one y and XYZ becomes yes and then uh between Alpha and beta there's like an x uh symbol and yes here yes and I named like top and bottom D left and right a and that makes 360 this D this d a a yes and I drew aord you're making it so complicated yeah but it doesn't it doesn't matter if you can manage to do it could you manage it no I'm trying to okay so you are sending me to the help okay okay no I thought that you solved that okay if you haven't solved that you still work on it okay that's a different problem but let me let me call this point e if you in case you want to name it uh okay so if you want this is I can call this x I can call this y I don't know if they are necessary let let us not mess it up we just name them if you really need them but you have to have a strategy to find uh the relation between Alpha and beta I want you to think a little bit yourself a lot of names here because you understand what is the goal the goal is to use the isos triangle idea because you have a little bit of experience from two simple cases in both cases we use the idea of an isoc triangle so try don't give up unless there is a good reason for it to give up so I don't know where is my green pen uh I don't know for example uh so this is one of your isoceles triangles you can imagine I don't know there is another isos triangle you can imagine the bigger one so let me call the center C that is Al the bigger one do you think that they are enough to consider what you need yes okay so think about it for example let us concentrate on this do you agree this is an isoc triangle why because this leg has length R this leg has also length R so if I call this GMA am I allowed to call this G as well it is a better one because you see I am not bringing two names in I'm bringing one name in usually in Geometry this is much better to do don't increase so you see say a a b b that's also working but that would be two letters but here we have one okay any other scenario is there any other one that you can see it's very simple you see so think about this angle can you give this angle a name of course you can give it ZX or whatever but tell me what can I write for this one GMA plus beta yes do you agree because this triangle is also what isoceles why because this is a steel R this one is a steel the radius so the length here and the length here are both equal to the radius so they are equal and if I ask you what is this angle it will tell me it is Alpha plus beta so then what does it mean for this angle Alpha plus sorry beta plus GMA that's also correct okay try to finish the problem but don't lose your goal your goal is to find the connection between Alpha and beta we haven't found that yet but I don't think it is hard to find it out could you you are waiting for me or you thinking is G okay so what you are doing you are considering this this triangle yeah and then if I call this X X plus this angle which is beta plus GMA plus this angle which is again beta plus GMA is 180 yeah so then I would like basic I WR 2 Beta plus 2 gam - 180 and then I prove that like Alpha is 2 GMA so this is what I agree with then let us write something else in this bigger one so then it becomes Alpha plus X Plus GMA plus GMA yes is again what 18 yeah and this is 100 180 this is also 180 so they have to be equal so this means that X Plus beta plus GMA plus beta plus GMA is equal to what Alpha + x + GMA plus G but then you see this comma move to the other side will cancel this comma this comma move to the other side cancel this comma X from this side move to the other side cancel that X and then what is left beta two beta what is left on the right that's what's good okay I'm not saying that this is the only way might be it's a little bit more complicated than usual you can find a little bit simpler way but I hope that you understand that in in all possible cases if I have one inscribed angle and if I have one central angle they are subtending the same Arc the size of which one is two times of the other one the size of the okay so I don't I hope that so this is something for the people who want to understand the reasons but what I expect you to in the exam is to know this rule tool and to be able to use it to solve problems so if I want to re summarize all all what we did we can momentarily forget about if you don't like this but you need to remember this if I have two angles in one Circle one of them inscribed the other one Central and they are subtending the same Arc the size of the central one is two times larger than the size of the inscribed one that is the message of what we have done so far is that clear yes or no okay so let me ask you one question if you understood it correctly I draw a circle here for you yes and then I draw a diameter any diameter that you like and then I pick up a random point a complete random point on the circle and then what I do I take these points of the diameter and connect it here here okay is there something I can say about this angle something completely concrete and definite or you might say that no it depends to where I choose the random point if I choose the random Point here this angle might be different if I choose the angle here you might say that this angle might be different can I say something about all these question mark angles first of all do you see that they have to be all equal let us understand that because that's very important you see if I give you this scenario I draw this angle this is what type of angle INB but these two points I take them and connect them now here then I get another angle beta can you see why these Alpha and beta are equal why they should be equal be careful this is inscribed that is also inscribed but they are subtending the same Arc In the case that one of them is Central one of them is inscribed the central is two times larger but here I am telling you these two are the same size Alpha is equal to Beta why is that you don't need to go through the long process this is something that you need to know if I have two inscribed angle opposite to the same Arc the sizes are exactly equal why exactly so I because I intentionally wanted to uh a little bit make it harder for yourselves let us say that this is Center I connect this Center to here and here then I generate an angle here GMA yes this GMA is a central angle and this angle is an inscribed angle but the central one subtains this Arc the inscribed one also subtains the same Arc so what is the relation between Alpha and GMA GMA is two times larger than Al yes this set it aside now forget about Alpha try to compare all gamma and beta according to the same idea what is the relation between beta and gamma because they are also subtending the same Arc one of them is Central one of them is inscribed so what can I say between GMA and beta GMA is again two times beta but now GMA and GMA are the same so it means that I have to agree that 2 Alpha is equal to 2 Beta And if 2 Alpha is equal to the beta and then it means that Alpha is equal to Beta in the exam you don't need to give reasons for that you can use them but you need to understand how to use them if you have two or more I mean for example well this picture is good to have in mind if I have this picture I don't know I take these two points and connect it here I take these two points and connect it for example here in this extreme if this angle is Alpha this angle is beta this angle is gamma what can I say about alpha beta and gamma they are equal okay because all of them are inscribed opposite to the same Arc if that is the case all of them inscribed opposite to the same Arc the sizes are the same if someone asks you why you should be able to explain how do you explain you would say that I take the center and then I take these two points and connect it to the center then this makes me an angle Theta Al Theta is 2 * Alpha because of central inscribed angle Theta is 2 * beta theta is 2 * gamma if 2 * Alpha 2 * beta 2 * GMA are all equal to Theta then it means Alpha Beta And GMA have to be equal themselves so this is I want you to understand not just to memorize but what is important for the exam is to be able to use the rule okay so now let us go back to this question I hope that now at least you understand that all question marks are equal can you tell me why why the size of this question mark and that question mark and that question mark are all the same do you remember this was diameter it is passing through the center why these question marks let me call them alpha beta and gamma again do you see why Alpha Beta And GMA are equal why all of them are what type of angles inscribed they are opposite to the same Arc what is that Arc this Arc all of them are inscribed opposite to the this Big Arc but that's Arc so it means that the sides of all these angles are the same Alpha Beta And GMA are the same is that understandable but in this case I can say something more not only they are equal but I exactly know what the value is yes what how is that possible that Alpha is 360 Yes refer to your brain a little beit yes is that possible no okay so nothing yeah of course you're right but then what is that size no don't say my picture is awful don't rely on my picture try to understand the logic so I hope that everyone do you everyone understands the connection of my explanation here and the connection here they are all inscribed they are all inscribed angles opposite to the same AR so the sizes according to this should be all the same but here if you ask me B do you know the sizes of these angles no I cannot say anything but I know that the sizes are equal but in this case I am saying that not only they are equal but I know exactly the size can you def yes that's 90 and why that is 9 because the central angle a yes where is the central angle it's a little bit hidden this is a central angle and everyone knows the size of the central angle what is the size of the central angle it's here 180 why because it's a diameter okay so this is a central angle subtending this Arc and these are inscribed angles subtending the same Arc according to the according to Central inscribed angle the size of the central one should be what 2 * larger than the size of the inscribed one if the size of the central one is 180 then the size of the inqu one should be 90 is that right so that is also an so if this happens in the exam you can use it what how should you refer it in the exam you say that inscribed angle opposite to a diameter so remember an inscrip angle opposite to a diameter is always 90 degrees it doesn't matter if the circle is big or small whatever and where the diameter is it doesn't matter so for example let me just try to draw another picture so I just draw another random one this is the center I draw a random diameter and then I pick a random point doesn't matter on the right or on the left and I connect it to the end of the diameter then what is this angle what is that angle what is that angle what 0 yes it is 90 yes why because it is an inscribed angle opposite to the diameter if you want to use it in the exam you don't need to prove it for me every time you need to tell me that okay that angle is an inscrib angle opposite to a diameter so it is 90° yes is that understandable okay now there is one interesting question here that is also in the book I want you to discover it yourself what I do I draw a circle here now any Circle but then I try to draw any quadril Al in a way quadrilateral is a shape with four sides so that the vertices are all on the circle okay so for example I take a point I take another point I take another point and I take another Point randomly okay and then I draw my quadrilateral if you want the name the name is not in the book but if you know an English name this is called a cyclic uh quadrilateral quadrilateral yes forget about the name okay what I want to tell you is that this is Alpha okay by the way let us review something from mat 1 C if this is Alpha this is beta this is GMA and this is Theta we learned something about Alpha plus beta plus GMA plus Theta not only in the circle in any quadrilateral we learned something about the sum what was that 36 360° do you remember more or less why because every quadrilateral can be partitioned into two triangles and for each triangle the sum is 180 so for the whole it would be 360° but if that quadrilateral is like this within a circle with the vertices on the circle we can say something interesting Alpha I don't know what the size is beta I don't know what the size is but I know something about Alpha plus beta I want you to tell me what that number is and why it doesn't matter how I draw the opposite ones the sum of the opposite ones is always one number if my quadrilateral lies on the circle if it is not no I cannot say anything more but if it is lying on the circle I can say Alpha plus beta plus GMA plus Theta is still 360° but in this case Alpha plus beta is one particular number always and what that number is okay can you defend that that's correct this becomes 180 okay can you tell me if this is the case what is left what is the share for the sum of these two then same the same because the sum should be 360° if you can defend this one it is automatically concluded because the sum of all of the is supposed to be 360° if you can defend this why this is true then of course this will follow immediately can you defend that I put a point exactly thank you we put we find the center there is somewhere in the center is somewhere and then what do you do line let me call them yes uh a to c is not good if you want to prove this because if I draw a to c I am splitting Alpha into two parts I I need the whole Alpha I mean I mean yes so I draw this line and I draw that line do you agree okay now my question for you is this if you don't mind let me call this angle X okay what is the name of this angle in general what type of angle is X what type of angle is that Central why because the vertex is the center and the legs are radi so what is the name of this angle what type of angle Alpha is what type of that what it's inscribed because the vertex is on the circle and the legs are quarts but do you think that Alpha and X are subtending the same Arc yes or no yeah yes because this x is subtending this Arc yes this is one leg this is another leg so this x is subtending this Arc what Arc is being subtended by Alpha do you see that it's exactly the same Arc yes so one of them Central one of them inscribed subtending the same Arc what is the conclusion between the size of X and the size of alpha size of X is what 2 * size of Al yes or no yes okay now let us visualize the problem from a different perspective if I call this Y what type of angle is why central yes because the Cent the vertex is the Center and the legs are radi what type of angle is beta inscribed why because the vertex is on the circle and the legs are cords but can I say beta and Y both subtain the same Arc yes which Arc this green Arc do you agree so the The Arc opposite to Y is this green part The Arc opposite to Beta is also the same thing so I have one central angle one inscribed angle subtending the same Arc what can I conclude about the sizes Y is equal to what to Beta yes but I my question was something related to Alpha plus beta I want to know something about Alpha plus beta so it motivates me to add them up x + y becomes equal 2 Alpha + 2 Beta I can Factor the two out yes it becomes Alpha plus beta agree or not do I know something about x no do I know something about Y no but do I know something about x + y 360° yes so it becomes 360° is equal to Al 2 Alpha plus beta so I looking for Alpha plus beta so what's Alpha plus beta 180° so this is also in your book I want you to understand that so if I have a random quadrilateral be careful if I have a random quadrilateral if this is Alpha if this is beta this is GMA this is Theta what can I write for this random one only one thing Alpha plus beta plus GMA plus Theta is what nothing more I can not conclude that Alpha plus beta is 180 GMA plus Theta is also sorry theta plus GMA is also no this is the only thing I can write but if it happens that a circle goes around that in a way that all vertices are on the circle then not only Alpha plus beta plus GMA plus Theta is 360° but I can say something more I can say the sum of the opposite ones are individually equal to 1 80 which I couldn't say in mat 1 C if I don't know if it's on the circle okay is that clear okay so that's also something I want you to know so let me see if there is something missing from the lesson that I haven't done so you have to train your eyes to catch this pattern okay and there are a lot of exercises in the book so if you don't mind let us try to show and solve some of them okay uh so let me see if it is being recorded or not no I don't know yes it is good okay so here determine these angles for example everyone everyone tell me what is this angle what is the size of angle x 57 57 yes what is the reason both of them are inscribed opposite to the same Arc so that's it can you tell me what is V here what is the size of v why because one of them is inscribe the other one is Central the central is two times larger than the inscribed so if it is the if the bigger one is 140° 14° then the other one will be half of it which is 57° is that understandable okay so for example can you tell me what is x y z in this picture all of them are 55 because those three are inscribed ones opposite to the same Arc and this Arc is subtended by a central angle of size 110 so all those inscribed ones should be half of it understandable by everyone everyone understands that so it is 55 yes okay so for example in this picture what is the size of U what is the size of v and u do you see that I can make it a little bit bigger so this one which one is 40 both of them but V is 40 based on today's lesson this one is inscribed this one is Central the central is two times larger so the inscribe is 40 but why U is 40 it has nothing to do with this lesson it is something related to math 1 C lesson why U is also 40 because this triangle that you see here here is an isoceles triangle why because both of them has the size of a radius is that right okay so this is you see it is not hard at this level okay so let us make it a little bit more interesting okay so can you tell me I want to test your understanding so can you tell me immediately this point is a center you know this point that you see you might not see I don't know from that distance this is a center the center what is U 90 what do you write in the exam if I ask you why you say that inscribed opposite to a diameter so this is this is inscribed opposite to a diameter so what is the size of V then 90 again for the same reason inscribed opposite to the diameter yes okay a little bit more interesting question is this one can you zoom yes I will try to see if it works yes I think it works so here this angle is 24 this angle is 53 and then they have put a point here of course they mean that is the center might be it is necessary we don't know yet and they are asking about u v and U and I would say that the point the center is not necessary at all it's like a butterfly so remember this shape of a butterfly in different directions okay so can you tell me everyone concentrate on V first can you tell me what is V 24 why is that because if I ask you what is this angle you will tell me it is inscribed if I ask you what is this angle you will tell me that is also inscribed but are they subtending the same Arc yes this Arc that you see here let me change it this Arc that you see here is subtended by this one and that one so they have the same sizes and what is the size of U for the same reason it is also 53 is that right okay so now let us see if you have a little bit more interesting questions than this okay so I want to wait for you okay to try to find this and then I will ask you okay question 4255 425 05 okay so here they are asking you about uh determining the angle p and the angle Q of course one of them is mentioned to be 3 a one of them is mentioned to be seven a I don't know what it has to do with this lesson I don't understand that it's a mat 1C lesson to be honest so what what is what is the size of angle p and what is the size of angle Q if that is Q I don't know where Q is yeah Q is here yes that's the center so what is that and they have mentioned you that this angle is 90 so I don't know what has what they want to do here what they want to do you know so you shouldn't get confused because you they have drawn a circle for you we don't care about that what we care is that they have given you this they have given you this angle is 90 okay and they have given you that this angle is uh 7 a and 3 a and they want you to tell me what is this angle and what is this angle so in order to answer this question you need to know the unknown a but how can I find the unknown a it has nothing to do with today's lesson how can I find that angle a yes because if this is 90 this one is also 90 and then I know that in any triangle the sum of the angles should be 180 so I move this to the other side and I add them up it becomes 10 a is equal to 90° I am looking for a so a becomes 9° but a is 9° they want you to find angle P so what is angle P angle p is 3 * a it is 27° what is angle Q angle Q is 7 * a which is 63 I don't know this has nothing to do with this okay but this is a little bit more interesting question question number 420 six yes 4206 I want to wait for you okay everyone solve this problem individually yes I can make it a little bit bigger and then move it a little bit here okay I want to wait for you to do this problem it's a little bit ambiguous you see this point is not clear is it on this line or not don't consider it on the line yeah I don't know why they have put the thought there the reason is clear they want to confuse you but why I don't know ah this one this one is easy do the part B this is not ambiguous at all this is a good question e-level question for the exam Part B solve it can you tell me what is good for this problem it's on the board still no this one I told you that if you have a quadrilateral without any circles you cannot say something more except this but if you know that the vertices on one given Circle you can say that the sum of opposite ones are 180 okay so if I go back to this uh if I go back here this one what so let me change this this one plus this one is supposed to be 180 okay so I I'll go to the board what is the angle one of them is 4X the other one is 8x it is supposed to be 180 this becomes 12x is 180 I divide everything by 12 to get X so X becomes 15° 180 divid by 12 is 15 yes but I want to find u and v where is V by the way in this picture we don't have any V we have only U I also know something else I know that this is I know that this one plus u should be again what 180 yes so it becomes 7 x u + 7 x is 180 what I found X from the previous calculation so I put it there 7 * 15° is what5 yes it becomes 180 then I am looking for U so U becomes 180 minus 105 becomes 75° yes so we were able to find you very easily so you see this is e- level question not hard you see even for this case it doesn't matter if this point is given to you or not forget about this point you have a quadrilateral you see that the vertices are where you yes the vertices are on a circle so regardless of this fact that if this is the center if the center lies on this line or not we don't care we have a quadrilateral the vertices of the quadrilateral is on a circle so what can we conclude we can conclude 63 plus v is 180 so what is the share for V 117 and then we also know that U + 121 should also be again 180 so regardless of this is a center or where the center is we don't care even in this case we don't care okay so U plus 121 is also 180 so what is the share for U uh 509 that's correct okay so we stopped the lesson here but before we leave I check the attendance