Proofs concerning isosceles triangles

So we're starting off with triangle ABC here. And we see from the drawing that we already know that the length of AB is equal to the length of AC, or line segment AB 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 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 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 ABC, I want to prove that that is congruent to angle ACB. 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 or the legs of the isosceles triangle. And these are-- obviously they're 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 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 congruency. 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 congruency, 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 distance-- 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 segment AD. And what's useful about that is that 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. So we know that triangle ABD we know that it is congruent 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 are 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 is angle ACD in this triangle right over here. So that we then know that angle ABC is congruent to angle ACB. 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. Let me 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. Let me do that 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 they have the same exact measure. And what we want to do in this case-- we want to prove-- so let me draw a little line here to show that we're doing a different idea. Here we're saying if these two sides are the same, then the base angles are going to be the same. We've proved 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 that segment AC is congruent to AB. Or you could say that the length of segment AC, which we would denote that way, is equal to the length of segment AB. These are essentially equivalent statements. So let's see. Once again in our toolkit, we have our congruency 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 as the midpoint, 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 you view BC as straight horizontal, the point that goes straight down from A. And the reason why I say that is there's some point-- you could call it an altitude-- that intersects BC at a right angle. And 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? And let me write this down. So I've constructed AD such that AD is perpendicular to BC. And you can always construct an altitude. Essentially, you just have to make BC lie flat on the ground. And then you just have to drop something from A, and that will give you point D. You can always do that with a triangle like this. So what does this give us? So over here, we have an angle, 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 is a valid congruent postulate. So we can say now that triangle ABD is congruent to triangle ACD. And we know that by angle-angle-side. This angle and this angle and this side. This angle and this angle and this side. And once we know these two triangles are congruent, we know that every corresponding angle or side of the two triangles are also going to be congruent. So then we know that AB is a corresponding side to AC. So these two sides must be congruent. And so you get AB is going to be congruent to AC, 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. If the two legs are equal, then the base angles are equal. It's a 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 D so it was directly below A. We didn't say whether it was the midpoint. But here, we can actually show that it is the midpoint just as a little bit of a bonus result, because we know that since these two triangles are congruent, BD is going to be congruent to DC because they are the corresponding sides. So it actually turns out that point D for an isosceles triangle, not only is it the midpoint but it is the place where, it is the point at which AD-- or we could say that AD is a perpendicular bisector of BC. So not only is AD perpendicular to BC, but it bisects it. That D is the midpoint of that entire base.