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Proofs concerning isosceles triangles

CCSS.Math:

Video transcript

so we're starting off with triangle ABC here and we see from the drawing that we already know that the length of a B is equal to the length of AC or line segment a B is congruent to line segment AC and since this is a triangle and two sides of this triangle are congruent or they have the same length we can say that this is an isosceles triangle isosceles isosceles triangle one of the hardest words for me to spell I think I got it right and that just means that two of the sides are equal to each other now what I want to do in this video is show what I want to prove so what I want to prove here is that these two and they're sometimes referred to as base angles these angles that are that are between one of the sides and the side that isn't necessarily equal to it and the other side that is equal and the side that's not equal to it I want to show that they're congruent so I want to prove that angle a b c proved that angle a b c i want to prove that that is congruent to angle a c b angle a CB and so for an isosceles triangle those two angles are often called base angles and this might be called the vertex angle over here and these are often called the sides and these are or the legs of the isosceles triangle and these are obviously their sides these are the legs of the isosceles triangle and this one down here that isn't necessarily the same as the other two you would call the base so let's see if we can prove that so we there's not a lot of information here just that these two sides are equal but we have in our toolkit a lot that we know about triangle congruence so maybe we can construct two triangles here that are congruent and then we can use that information to figure out whether this angle is congruent to that angle there and the first step if we're going to use triangle congruence is to actually construct two triangles so one way to construct two triangles is let's set up another point right over here let's set up another point D and let's just say that D is the midpoint of B and C so it's the midpoint so the distance from B to D is going to be the same thing as the since let me do a double slash here to show you it's not the same as that distance so the distance from B to D is going to be the same thing as the distance from D to C and obviously between any two points you have a midpoint and so let me draw let me draw a segment ad let me draw a segment ad and what's useful about that is that we've now constructed we have now constructed two triangles and what's even cooler is that triangle abd and triangle ACD they have this side is congruent this side is congruent and they actually share this side right over here they actually share that side right over there so we know that triangle we know that triangle abd triangle a BD we know that it is congruent to triangle ACD to triangle ACD and we know it because of SSS side side side you have two triangles that have three sides that are congruent or they have the same length then the two triangles aren't congruent and what's useful about that is if these two triangles are congruent then their corresponding angles are congruent and so we've actually now proved our result because the corresponding angle to ABC in this triangle the corresponding angle to this is angle ACD in this triangle right over here so that we then know that angle ABC is congruent is congruent to angle to angle a/c B so that's a pretty neat result if you have an isosceles triangle a triangle where two of the sides are congruent then their base angles these base angles are also going to be congruent now let's think about it the other way can we make the other statement if the base angles are congruent do we know that these two legs are going to be congruent so let's try to construct a triangle and see if we can prove it the other way so I'll do another triangle right over here we draw another one just like that that's not that pretty of a triangle so let me draw it a little nicer I'm going to draw it like this I'm going to call this one let me do it in a different color so I'll call that a I will call this B I will call that C right over there and now we're going to start off with the idea that this angle angle ABC is congruent to angle ACB so this is where they have the same exact measure and what we want to do in this case we want to prove so we draw a little line here to show that we're doing a different a different idea here we're saying well if these two sides are the same then the base angles are going to be the same we prove that now let's go the other way if the base angles are the same do we know that the two sides are the same so we want to prove we want to prove that segment a B is congruent to segment AC or AC is congruent to a B to a B or you could say that this length of segment AC which we would denote that way is equal to the length is equal to the length of segment a B these are essentially equivalent statements so let's see once again in our tool kit we have our congruence theorems but in order to apply them you really do need to have two triangles so let's construct two triangles here and this time instead of defining another point at the midpoint I'm going to define I'm going to define D I'm going to define D this time as the point that if I were to go straight up the point that is essentially if if we view BC is straight horizontal if you the point that goes straight down from a and the reason why I say that is there's some point there's some point you could call it an altitude that intersects BC at a right angle and there there will definitely be some point like that and so if it's a right angle on that side if that's 90 degrees then we know that this is 90 degrees as well now what's interesting about this well let me write this down so I've constructed constructed I've constructed ad ad such that such that ad ad is perpendicular to BC is perpendicular to BC and you can always construct an altitude essentially you just have to make ABC lie flat on the ground and then you just have to drop something from a and that'll give you point D you can always do that with a triangle like this so let's so what does this give us so over here we have an angle at an angle and then a side in common and over here you have an angle that corresponds to that angle an angle that corresponds to this angle and the same side in common and so we know that these triangles are congruent by AAS angle angle side which we've shown as a is a valid congruent postulate so we can say now that triangle triangle abd a B abd is congruent to triangle ACD a ACD and we know that by angle angle side this angle then this angle on this side this angle and this angle is on this side and once we know these two triangles congruent we know that every every corresponding angle or side of the two of the two triangles are also going to be congruent so then we know that a a B is a corresponding side to a C so these two sides these two sides must be congruent and so you get a B you get a B is going to be congruent is going to be congruent to a C and that's because these are congruent triangles and we've proven what we wanted to show if the base angles are equal then the two legs are going to be equal the two legs are equal then the base angles are equal very very very useful tool in geometry and in case you're curious for this specific isosceles triangle over here we set up D so it was the midpoint over here we set up Diesel 20 below a we didn't say that whether it was a midpoint but here we can actually show that it is the midpoint just is a little bit of a bonus result because we know that since these two triangles are congruent BD is going to be VD is going to be congruent to DC because they are the corresponding side so it actually turns out that point D for an isosceles triangle not only is it the midpoint not only is it the midpoint but it is the place where it is a point at which at which ad or we could say that ad is a perpendicular bisector of B of BC so not only does it not only is ad perpendicular to BC but it bisects it that D is the midpoint of that entire base