Geometry

# Triangle Angle Sum

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!