 In this video, we wanna discuss a very cherished truth from geometry, and that is commonly known as the angle sum of a triangle theorem. That is to say that if you have a triangle, like I drew a picture of the triangle on the screen right here, this ABC triangle, the sum of the three angle measures of every triangle is gonna add up to 180 degrees. So what we see here is that I take the measure of angle A and I add that to the measure of angle B and I add that to the measure of angle C. This always adds up to 180 degrees and this is true for every triangle in our Euclidean geometry here. That is to say the sum of three angles forms a line. They're gonna be supplementary to each other. That is when you add them all together. And the typical argument here actually comes from this idea that we wanna show that the sum of these three angles forms a half angle. That is a forms a half plane here. So what I want you to do is consider this triangle and we're going to add a parallel line to the line AB on the bottom here. So you could like extend AB if you wanted to. So we have this line AB, we're gonna draw this new line L and what we wanna say here is that the line determined by A and B is parallel to this new line L. We also want that the point C belongs to the line L. So we construct a picture like this. And so now I want you to consider angle A for a moment, okay? AB as a line and L are parallel lines. And so the line AC acts as a transversal to these set lines. And so by the alternative angle theorem, angle A is gonna be congruent to this angle right here. So we'll call this one angle one. So angle A is congruent to angle one. But also by the alternative angle theorem, if we consider the line BC as a transversal to these two parallel lines, angle B, it's alternate interior angle will be right here. Let's call this angle two. So by the alternate interior angle theorem, you get that angle B is congruent to this angle two, okay? And so then the idea here coming back to that, we wanna consider the sum of the three angles, A plus B plus C. Well, by the observation we just made here, angle A is the same thing as angle one. Angle B is the same as angle two. And then angle C, of course, is the same thing as itself. But if you take these angles one, two, and C, those three angles come together to form the half plane associated to L. So these are supplementary angles, and therefore this adds up to 180 degrees. So as a consequence of the alternate to your angle theorem, we see that every triangle has an angle sum of 180 degrees. Let's see an example of such. If the measure of two of the angles of a triangle are given as 48 degrees and 61 degrees, what is the measure of the third angle? This is a calculation we do in trigonometry all the time. So what's the measure of angle X right here? Well, we have that 48 degrees plus 61 degrees plus X degrees is equal to 180 degrees. So to solve for X, we just take 180 and we subtract from it 48 degrees and we have to subtract from it 61 degrees, which is a fairly simple chore of arithmetic here. 108 degrees take away 48 degrees would be 132 degrees. Then if we take away 61 degrees, we end up with 71 degrees. And so that missing angle must have been 71 degrees. And we can figure that out because the angle sum adds up to 180 degrees each and every time. So because of the angle sum theorem, a triangle cannot have two obtuse angles because if you have two angles bigger than 90 degrees, like if you had 91 degrees plus 100 degrees, if you had two obtuse angles, that would add to be 191 degrees, which not even considering what the other angle turned out to be, let's take the measure of angle A or something like that, the measure of angle A. This is already bigger than 180 degrees, which is a contradiction. So no triangle can have two obtuse angles. By similar reasoning, no triangle can have two right angles and no triangle could have a right angle and an obtuse angle. And for this reason, we often label triangles based upon their largest angle. In this example, you'll notice the largest angle present is a 71 degree angle to an acute angle. This is an example of an acute triangle. Right triangles, which we've talked about previously in this lecture series, and we'll talk about extensively in this lecture series in future videos, a right triangle is exactly a triangle with one right angle. And similarly, an obtuse triangle is a triangle with exactly one obtuse angle. And so we can classify triangles based upon their largest angle as a consequence of the angle sum theorem.