I believe in asking “why.” So for my first real post(!), I want to talk about one fundamental theorem in geometry, Triangle Angle Sum, and *how* this theorem actually works. Hopefully by seeing an explanation of Triangle Angle Sum, you’ll better remember ideas that come from it.

So let’s get started!

First, you’ll need a triangle, of course. Here’s mine:

The Triangle Angle Sum theorem states that the sum of the angles in any triangle equals 180°. We know that angles with a sum of 180° can also be called **supplementary**. So we’re eventually going to show that the angles in a triangle can be rearranged to form a straight angle, which is the angle formed by a straight line.

The first step in rearranging our angles is to pick a base, such as AC in my drawing. Then draw a line parallel to the base that passes through the remaining point (for me, point B). Here’s what it looks like so far:

Let’s rearrange our angles around the parallel line DE:

- Angles BAC and ABE are congruent because of alternate interior angles;
- angles BCA and CBD are congruent because of alternate interior angles; and
- angle ABC is already part of the straight angle.

After labeling the angles, the triangle looks like this:

And voila! we have three supplementary angles that are also the angles of our triangle. Therefore the sum of the angles in a triangle (Triangle Angle Sum) is 180°.

**To summarize,** the sum of the angles in a triangle is 180° because the angles can be rearranged to form a straight angle.

If you have any questions or comments, feel free to leave a comment below. And if you learned anything from this post, don’t forget to share it with the world!

Thanks!

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