Theorems concerning triangle properties
What we've got over here is a triangle where all three sides have the same length, or all three sides are congruent to each other. And a triangle like this we call equilateral. This is an equilateral triangle. Now what I want to do is prove that if all three sides are the same, then we know that all three angles are going to have the same measure. So let's think how we can do this. Well, first of all, we could just look at-- we know that AB is equal to AC. So let's just pretend that we don't even know that this also happens to be equal to BC. And we know for isosceles triangles, if two legs have the same length, then the base angles have the same length. So let's write this down. We know that angle ABC is going to be congruent to angle ACD. So let me write this down. We know angle ABC is congruent to angle ACB. So maybe this is my statement right over here. And then we have reason. And the reason here, and I'll write it in just kind of shorthand, is that they're base angles of, I guess you could say an isosceles. Because we know that this side is equal to that side. And obviously, this is an equilateral. All of the sides are equal. But the fact that these two legs are equal so that the base angles are equal. So we say two legs equal imply base angles are going to be equal. And that just comes from what we actually did in the last video with isosceles triangles. But we can also view this triangle the other way. We could also say that maybe this angle over here is the vertex angle, and maybe these two are the base angles. Because then you have a situation where this side and this side are congruent to each other. And then that angle and that angle are going to the base angles. So you could say angle CAB is going to be congruent to angle ABC, really for the same reason. We're now looking at different legs here and different base angles. This would now be the base in this example. You can imagine turning an isosceles triangle on its side. But it's the exact same logic. So let's just review what I talked about. These two sides are equal, which imply these two base angles are equal. These two sides being equal implied these two base angles are equal. Well, if ABC is congruent to ACD and is congruent to CAB, then all of these angles are congruent to each other. So then we get angle ABC is congruent to angle ACB, which is congruent to angle CAB. And that pretty much gives us all of the angles. So if you have an equilateral triangle, it's actually an equiangular triangle as well. All of the angles are going to be the same. And you actually know what that measure is. If you have three things that are the same-- so let's call that x, x, x-- and they add up to 180, you get x plus x plus x is equal to 180, or 3x is equal to 180. Divide both sides by 3, you get x is equal to 60 degrees. So in an equilateral triangle, not only are they all the same angles, but they're all equal to exactly-- they're all 60 degree angles. Now let's think about it the other way around. Let's say I have a triangle. Let's say we've got ourselves a triangle where all of the angles are the same. So let's say that's point X, point Y, and point Z. And we know that all the angles are the same. So we know that this angle is congruent to this angle is congruent to that angle. So what we showed in the last video on isosceles triangles is that if two base angles are the same, then the corresponding legs are also going to be the same. So we know, for example, that YX is congruent to YZ. And we know that because the base angles are congruent. Now we also know that YZ-- so I'll rewrite YZ-- is congruent to XZ, by the same argument. But here we're dealing with different base angles. So now, once again, you can view this as almost an isosceles triangle turned on its side. This is the vertex angle right over here. These are the two base angles. This would be the base now. And we know that because these two base angles are congruent. So by the same logic. Over in this first case, the base angles were this angle and that angle. In the second case, the base angles are that angle and that angle. And actually let me write it down. The base angles in this first case-- let me do that same magenta-- are angle YXZ is congruent to angle YZX. That was in the first case. These are the base angles. So based on the proof we saw in the last video, that implies these sides are congruent. Here, we have these two base angles. Let me do that in green. Angle XYZ is congruent to angle YXZ. And so that implies that these two guys right over here are congruent. Well, there we've proved it. We've said that this side YX is congruent to YZ. And we've shown that YZ is congruent to XZ. So all of the sides are congruent to each other. So once again, if you have all the angles equal, and they're going to have to be 60 degrees, then you know that all of the sides are going to be equal as well. They're going to be congruent.