 Hello, and welcome back to Tutor Terrific. Today, we're going to show the proof of the triangle sum conjecture, also known as the triangle sum theorem. This theorem states that for any triangle, no matter what, if it is actually a triangle, that means a polygon with three sides, the sum of the interior angles is always 180 degrees in a Euclidean space. That means the planes are flat. They're not curved. So, this is one of the most famous proofs, most famous conjectures in all of geometry. And so, we should know how to prove it. And often, high school students are asked to do this proof. So, I know of one school, in particular, every test after they learn this conjecture for the first semester, they have to do this proof from memory. That's very important. So, here is how this proof is done. We create a triangle, and we label the sides, and we label the angles. So, here are all the labels. So, this would be side AC, this would be side CB, and this would be side AB. What I'm going to do is cleverly make a blind parallel to side AB. Now, who says I could do this? Well, something called the parallel postulate. If you have a compass, you can actually, and a straight edge, you can actually create a line that is parallel to a given line through a point not on the given line, and that is the parallel postulate. So, this proof has five steps, and the first step is to list that you can do that. I can create line EC parallel, this is the parallel symbol, to segment AB because of the parallel postulate. Okay, just the fact that I could do that is necessary, and so I actually construct that line. I didn't really construct it here, I sketched it, but it can be constructed. And it is parallel to line AB. Okay, now, I'm going to, by doing this, create two more angles. This angle I shall call angle four, and this one I shall call angle five. Now, let's look at all this stuff going around point C. Angle ECA, angle ACB, and angle BCG. These angles all add up to a straight line, these three angles. So, in other words, it's a term I coined a linear triple. So, you could say that the measure of angle four plus measure of angle two plus measure angle five equals 180 degrees. This is the backbone equation we use to prove our equation. Notice how similar it is. Now, for each step of the proof, you must say the reason you can do it. So, we say here that these three angles add to a straight line. Usually what's done is the word supplementary. These angles would be supplementary, but we usually refer to that only when we're talking about two angles adding up to 180. But, these are a linear triple. They add up to a straight line. So, their measures add up to 180. And that, by the way, is why I used the little M in front of the symbol for the object. Measure of is what that means. Step three involves these parallel lines and sides CA and CB. I want you to look at angle four and angle one together as a set. Notice how they are alternate interior angles to these two parallel lines in transversal CA. Well, because those lines are parallel, these angles are congruent. So, angle one is congruent. This is the congruent sign to angle four because... Oh, and we won't stop there. We could say another one too. Angle five and angle three do the exact same thing with transversal CB. So, angle three is congruent to angle five because... Now, I will say this in shorthand, so I can fit this all in one page, so you can really see that this is a very digestible proof. Alternate interior angles theorem. AIA stands for Alternate Interior Angle Theorem. So, these angles are congruent. Angle one congruent to angle four. Angle five congruent to angle three. It seems like angle two is the only odd one out in this case without an angle congruent to it, but that's okay. Now, step four, it's a subtle step, but it's absolutely necessary. If two angles are congruent, that means that the two measures of the angles are equal. That follows directly. So, measure angle one equals the measure of angle four and measure of angle three equals the measure of angle five. And we'll state the reason like this because congruent angles have equal measures. Congruent angles have equal measures. See how I went from the congruent statement to the measures are equal statement. And now it's time for the final step. Step five, we are going to do a simple substitution. Let's refer back to step two, where I said measure angle four plus measure angle two plus measure angle five equals 180. Well, step four said that measure angle one equals measure angle four. So, I could substitute that in. So, and I have measure angle one plus measure angle two. Now, before I write measure angle five, I know that measure angle three equals measure angle five. So, I could substitute measure angle three in right there. And this all apparently equals 180 degrees. So, this is because of the substitution property. What I just did was substitution, substitution property. As you can see, we have the result. We desired the first place and we are finished with the proof of the triangle sum conjecture. Thank you so much for watching. And for now, this is Falconator signing out.