Why is the sum 180 degrees?

Added: 2026-05-30

[00:00:04]If you add up the three angles of a triangle, you always end up with that 180°.

[00:00:11]But I'm sure you need this already. But have you ever asked yourself why it's 180° each and every time? In this video, I'm going to show you using a very important theorem why when you add up three angles of a triangle, they add up to 180°. And then at the end of the video, as a bonus, I'm going to be using this piece of paper to show you a second method on why the three angles add up to 180 degrees. Let's start by drawing these two lines on the top and the bottom of the triangle and then label them X and Y. And we're also going to write that the line X and also the line Y are parallel to each other. And in geometry, we mark the lines with these two arrows to show that the lines are parallel to each other. Then the theorem that we're going to be using is the alternate interior angle theorem.

[00:01:07]What alternate interior angle theorem says that is if we have two lines that are parallel to each other, then the angle here and on the opposite side, notice they're both inside the parallel lines. That's why they're called in interior. And because they're alternate of each other that these two angles are congruent to each other or equal to each other using this if we have angle A right here then alternate interior angle or on the other side of these this angle here is going to be a degrees also. Likewise, if we have angle C here, then the alternate interior angle or right there is going to be angle C as well. Then another thing that I want to mention is this. If you have a line like so, then it's called a straight angle. And straight angle is always going to be 180°.

[00:02:08]Then no matter how many angles it's broken up into, the angles will always add up to 180°.

[00:02:18]That's what we have here. That line X from here to here is going to be 180°. Meaning if I take the angles A, B, and C, they add up to 180°. Hence we get the formula or the theorem saying that the ABC or the three angles of a triangle always add up to 180°. There you go folks. Now you know why the three angles of a triangle is always 180°. So you don't have to wonder about it anymore. Now let me show you a second method which is even more simple and quick uh than this method to show why the three angles of a triangle at 180°.

[00:03:08]And now for the second method on showing that three angles of a triangle at 180° using this piece of paper. Notice I marked the angles A, B, and C. And I'm going to separate the three angles like this. First, let's go and put up angle A.

[00:03:28]on that straight line. And then the second angle B like that. And then finally the third angle C like this. Notice the three angles add up to 180 as you can see. So there you go folks. This is quite visually satisfying to see that when you take the three angles of a triangle and then line them up on a straight angle that they add up to 180°.

[00:04:02]So, what do you think? Which method do you like more, the first or the second?

[00:04:06]Although I do like the first method, I got to admit I like this method more. On that note, I'd like to thank my wife for showing me this method.