Transcript for:
Basic Concepts in Geometry

in this lesson i'm going to go over some basic concepts in geometry that you want to be familiar with so the first thing we're going to talk about is a line what is a line now many of you may think of a line as just something that looks like that but in geometry a line would have two arrows and so it extends in opposite directions forever now let's put some points on this line so we can call this line line a b we could also call it line bc and we could also call it line ac so there's many ways in which we can name this particular line now what about a ray how would you describe a ray array has a point and it extends forever in the other direction so it has a point and an arrow so let's put some other points on this array let's call this a b c so we can name this ray ac or we can name it ray not bc but ac i mean a b the reason why we can't name it ray bc is because the first point has to start at the initial point or the starting point of the ray so we need to use point a as our starting point now the next one you need to be familiar with is something called a segment a segment has a beginning and it has an end so let's say this is point a and point b so we can describe it as segment a b with no arrows on it or we could say this is segment the a so that's a segment now let's move on to angles there's four different angles that you need to be familiar with so the first one is an acute angle acute angles have an angle measure of anywhere between 0 and 90 but it's less than 90 that's what you want to take from this and so we can call this angle abc and if you notice an angle is formed by the the union of two rays so here we have ray ba and ray bc and so this forms at the vertex creating an angle the next angle is the right angle and a right angle has an angle measure of 90 degrees the next one you need to be familiar with is something called an obtuse angle so the obtuse angle has an angle that's greater than 90 but less than 180 and finally you need to be familiar with the straight angle the straight angle has a measure of 180 degrees so that's a straight angle it looks basically like a line now let's move on to the midpoint what is a midpoint well if you look at the word it sounds like the point in the middle and you wouldn't be incorrect by saying that so let's say if we have a segment we'll call it segment ac and let's say b is the midpoint so what does that mean that means that b is right in the middle of segment ac it also means that segment a b and b c are congruent so if you put these marks it shows that this segment is equal to that segment and the way you would write that is as follows so you would say segment a b is congruent to segment bc and so if that's true that means that b is the midpoint of segment ac thus b bisects segment ac into two equal parts now another term you need to be familiar with is something called what just happened here something called a segment bisector so let's say if we have a ray that passes through the midpoint and let's say this is r so ray rb is considered a segment bisector because it bisects the segment into two congruent parts so we can see that a b and b c are congruent to each other next up in our list of things to discuss is the angle bisector so what is an angle bisector an angle bisector is a ray that bisects an angle into two equal parts so let's say we have let's call this angle abc and we have a ray that emanates from point b and let's call this ray bd so ray bd is an angle bisector if it bisects angle abc into two equal parts so let's say that angle abc is 60 degrees by the way when you name an angle the vertex of the angle has to be in the middle so you can call it angle abc or you can call it angle cba so b has to be in the middle so if ray bd is an angle bisector that means that angle abd has to be 30 and angle dbc has to be 30. now let's move on to parallel lines so what do you know about parallel lines what are some properties of parallel lines parallel lines never intersect so let's call this line a and line b parallel lines also have the same slope so let's say the slope of line a is a half the slope of line b will also be a half and the way you can describe that these two lines are parallel the geometry symbol for it looks like this so you say a is parallel to b so you would use two lines to indicate that a is parallel to b now let's talk about perpendicular lines what are perpendicular lines and how are they different from parallel lines as we said before parallel lines do not intersect each other but perpendicular lines they do intersect each other in fact they do so at right angles so let's call this line a and line b so they're going to meet at a 90 degree angle now let's say that the slope of line b is positive two over five what is the slope of line a given that it's perpendicular to line b to find the slope of a perpendicular line first you need to flip the fraction and change the sign from positive to negative so the slopes are negative reciprocals of each other and the symbol for it goes as follows so we can say line a is perpendicular to line b so that's the geometric symbol that correlates to uh perpendicular lines the next term that you need to know in geometry is complementary angles so what are complementary angles complementary angles are angles that add up to 90 degrees so let's say if we have a right and let's call this triangle abc so the symbol for that is as follows so this is triangle abc and let's say that angle a is 40 degrees and angle c has a measure of 50 degrees in this case we could say that angle a and angle c are complementary because their angle measures add up to 90 degrees and so that's the basic idea behind complementary angles so the measure of angle a plus the measure of angle c is equal to 90. next we have supplementary angles now based on the last topic you can probably guess what this is about supplementary angles are angles that add up to 180. now if you remember a straight angle has an angle measure of 180 so we need to be dealing with a line so let's put some points let's say this is point a b c and d let's say angle abd has a measure of 110 and angle dbc has what measure what would you say it should be now remember a straight angle is 180 so 110 plus what number is 180. this has to be 70. so because the measure of angle abd and the measure of angle let's call this cbd or you could say dbc because they add up to 180 we could say that those two angles are supplementary angles the next geometric term that you need to be familiar with is something called the transitive property so what is the transitive property the definition is as follows if two angles are congruent to the same angle then they are congruent to each other okay so what does that mean let's say if angle one is congruent to angle two and if angle 3 is congruent to angle 2 then we can come up with this conclusion we could say that angle 1 is congruent to angle 3. now it seems obvious but you need to know that this property is it matches this situation because if you're doing let's say a problem with two column proofs you need to know that the transitive property corresponds to this particular situation now another way in which you could describe the transitive property is using variables so you can say that if a is equal to b and if c is equal to b then that means that a has to equal c so we can write then a is equal to c and that's the basic idea behind the transitive property so that's something else that you want to add to your notes next you need to be familiar with vertical angles if you're working on two column proofs and problems similar to that so what are vertical angles well we can form vertical angles by drawing two lines that intersect each other let's call this angle one two three and four so the opposite angles are congruent angle one is congruent to angle three so thus you can describe them as vertical angles angle two and angle four they're also vertical angles they are congruent to each other so let me give an example problem let's say if this angle is 50 what are the other three angles this angle has to be 50 because the vertical angles now these two angles let's call this angle x and 50 they form a linear pair so therefore they have to add up to 180. 180 minus 50 is 130 so the angle opposite to 130 is the vertical angle to it so this must also be 130 and so that's how you can use vertical angles to find missing angles in a typical geometry course now for those of you who haven't done so already feel free to subscribe to this channel and check out the description section of this video when you get a chance i'm going to post some links to it because in this video i don't have many practice problems and for those of you who want it i'm going to have another video with plenty of practice problems i'm also going to include my geometry video playlists for those of you who need help on specific topics and geometry so when you get a chance check out the description section of this video and don't forget to subscribe and if you decide to do that make sure to click on that notification bell so you can receive new videos that i'm going to post in the future now let's talk about medians what is a median to understand this concept let's begin by drawing a triangle and so let's call this triangle abc and we're going to draw a line segment from b to some point which we'll call d now in this example segment bd is a median a median is a line segment drawn from the vertex of a triangle to the midpoint on the other side of the triangle so that means that d is the midpoint so if d is the midpoint of segment ac that means a d and dc are congruent to each other and so that's what a median does the median is just a line segment it touches the midpoint of the other side and it splits one side of the triangle into two equal parts now let's talk about altitudes what is an altitude and how does it relate to a median so let's use another triangle as our example and we're going to call it triangle abc again with point d so this time bd is going to represent an altitude so like a median an altitude is a line segment but the difference is d is not a midpoint the altitude forms right triangles with in this case segment ac so angle adb which is congruent to angle c db both angles are equal to 90 degrees so the altitude forms right angles inside the triangle and that's what you need to know now we can also say that segment bd is perpendicular to segment ac because perpendicular lines and perpendicular segments they intersect at right angles so if you have an altitude you can also make this statement now the next thing we're going to talk about is a perpendicular bisector so looking at the words perpendicular and bisector what do you think this represents so let's say if we have a line segment which we'll call segment a b and we're going to draw a line that's perpendicular to it and let's call this line l so line l is perpendicular to segment a b which means that it's going to form a right angle with segment a b now it's also a bisector so if we draw another point we'll call it point m m is the midpoint of segment a b so a perpendicular bisector forms right angles with the segment that it intersects and at the same time it also bisects the segment into two congruent parts so when dealing with perpendicular bisectors you can say things like segment am is congruent to segment bm you can also say that m is a midpoint and you could also say that angle let's add some points here so let's call this point p and point q so we could say that angle amp and angle bmp these are right angles so perpendicular bisectors they are lines that bisect a segment into two equal parts and they form right angles with that segment personally i like to think of perpendicular bisectors as a hybrid of a median and an altitude so like a median it touches the segment at the midpoint and it bisects it into two equal parts and like an altitude it forms right angles so it has features of both now there's something else that you need to know about perpendicular bisectors and let's focus on point p so notice that am and bm they're congruent to each other and these two triangles they share a common side side pm therefore because these two sides are the same and this side is a common side which is associated with the reflexive property it turns out that ap is congruent to bp and so that's another feature of perpendicular bisectors is that any point on the perpendicular bisector is equally distant to the endpoints of the segment that it bisects so p is equidistant to points a and b so we can say that ap is congruent to bp and we could just we could also say the same of point q so aq and bq they're congruent to each other so those are some things that you want to keep in mind when dealing with perpendicular bisectors a common problem that you're going to encounter in geometry is proven that two triangles are congruent and so let's say if these are the two triangles we're dealing with we have triangle abc and triangle d e f now there's four postulates that you need to be familiar with there's some other ones too but we're going to cover four of them so let's say that a b and d e are congruent to each other and bc is congruent to ef and ac is congruent to df then we could say that triangle abc is congruent to triangle d e f according to the sss postulate because all three sides are congruent to each other and so that's the first postulate you need to be familiar with now when you write this expression you need to write it in such a way that the letters of each triangle correspond to each other so angle a corresponds to angle d because they're opposite to congruent sides likewise angle b corresponds to angle e they're opposite to congruent sides and so we need to make sure that they match when writing this statement and angle c corresponds to angle f now let's look at another example but let's use different letters so let's call this triangle x y z and let's call this triangle rst and let's say that segment x y is congruent to rs and segment xz is congruent to rt and angle x is congruent to angle r what partially can we use to prove that these two triangles are congruent so we have a side an angle and the side so we could use the sas postulate to make the statement that triangle x y z is congruent to triangle r s t and so that's the second postulate that you need to be familiar with in order to prove that two triangles are congruent now once you prove that two triangles are congruent you can also make another statement for instance now that we know that these two triangles are congruent i can say that angle y is congruent to angle s and the reason why i can say that is something called cpctc corresponding parts of congruent triangles are congruent so once you prove that two triangles are congruent then you can make a statement that other portions of that triangle are congruent as well using this as a reason now let's try another example so let's call this triangle m n o and this one we're going to call it g h i now let's say that angle n is congruent to angle h segment mn is congruent to segment gh and angle m is congruent to angle g so what postulate can we use to prove that these two triangles are congruent so we have an angle a side and an angle so we could use the a asa postulate angle side angle to say that triangle m n o is congruent to triangle g h i so now that we've proven that the two triangles are congruent i can make a statement and say that another portion of these triangles are congruent i could say that m o is congruent to segment gi and the reason is cpctc corresponding parts of congruent triangles are congruent now there's one more postulate that i want to go over so let's say this is triangle abc and this is triangle r as t let's say that angle a is congruent to angle r angle c is congruent to angle t and let's say that bc is congruent to st so what postulate can we use to prove that these two triangles are congruent so here we have an angle an angle and a side so we could use the aas postulate to make the statement that triangle abc is congruent to triangle rst and so that's the fourth postulate you want to know there's another one called the hypotenuse leg postulate which i'm not going to post in this video but you could find it if you check out my new geometry video playlist or if you type in h gel postulate organic chemistry tutor in your youtube search box it should come up now let's work on some problems so let's say that this is a d c and we're going to have a composite triangle so let's say that we're given that segment a d is congruent to segment dc and segment a b is congruent to segment bc how can we prove that these two triangles are congruent that is triangle abd and cbd what can we do right now we see that two sides are congruent but notice that both triangles share a congruent side if we were to split this triangle into two parts such that we have triangle a b d and triangle c b d so we can see that a d and d c are congruent because that's given to us a b is congruent to bc and also db is congruent to itself based on the reflexive property so that's we could say that triangle abd is congruent to triangle c bd according to the sss postulate now using cpctc we can make a statement that let's say angle a is congruent to angle c based on cpctc here is another example so let's say this is a b c d and e and let's say we're given that angle a is congruent to angle e and ac is congruent to ce prove that angles b and d are congruent now you could set this up as a two column proof but i'm not going to do that here i'm just going to discuss it the process of proving that these two triangles are congruent but if you want actual two column proof problems check out the links in the description of this video i'm gonna post some videos where you can find more practice on this topic this video is just a quick review of topics that you need to know in geometry so right now we can't use any postulates here because all we have is an angle and a side we don't have this yet this is what we're trying to prove so we can't use that right now so what else do we know about triangles acb and ecd notice that angles acb and ecb are congruent to each other because they form vertical angles vertical angles are congruent so based on the asa angle side angle postulate we can make the statement that triangle a c b is congruent to triangle e c d now once we prove that the two triangles are congruent now we can make the statement that angle b is congruent to angle d using c p c t c and we don't have to stop there if we want to we can show that segment a b is congruent to segment ed also using cpctc as a reason now let's consider one more example so let's say this is triangle abc and let's say that we're given that bd is an altitude in addition let's say we're given that angle a is congruent to angle c so let me just write given your task is to verbally prove that angle abd is congruent to angle c bd so feel free to pause the video and come up with a process that will prove that this statement is true so first let's mark down what we know angle a is congruent to angle c and so that's given now bd is an altitude what do we know about altitudes altitudes they form right angles so angle adb is congruent to angle cdb now in addition triangle a d b share a same side with triangle c b d so if we split it into two triangles triangle adb and cdb we could say that these two sides are congruent based on the reflexive property and we have that angle d is congruent to each other and angle a is congruent so thus we have angle angle side and so we can prove that the two triangles are congruent so we can make the statement that triangle a db is congruent to triangle c db now that we've proven that the two triangles are congruent we can make the statement that angle abd is congruent to angle cbd according to cpctc corresponding parts of congruent triangles are congruent and so that would complete this problem but you could set it up as a two column proof if you want to and using cpctc we can also say that segment a b is congruent to segment bc or we could say that segment a d is congruent to segment cd and so using cpctc you can also prove that any other parts of the two triangles are congruent so that's my two cents for today that's all i got and before you leave don't forget to subscribe to this channel and check out the links in the description for those of you who want more practice problems thanks again for watching