How Do Angles in a Triangle Add Up to 180 Degrees?

We've been taught that the three interior angles of a triangle always add up to 180 degrees. It's a fundamental property of triangles that's important in geometry and many other fields, including architecture, engineering and even art.

This well-known and widely accepted fact in geometry applies to ALL triangles, irrespective of their shape and size.

You may have memorized this handy math fact, but never really understood why it’s true.

After all, 180 seems like an arbitrary sum:

  • How did mathematicians discover this rule?

  • Is there an easy way to "see" the angles add up to 180 degrees?

Wonder no more!

Watch Muzology’s Mathematician-at-Large, Dr. James Tanton (founder of Global Math Project) show you how to literally "see" and understand why this is true...with the help of a magic marker!!

An angle is formed when two straight lines intersect at a point. The size of an angle is determined by the amount of rotation needed to align one of the lines with the other. This rotation is measured in degrees, with a full rotation around a point (360 degrees) being equivalent to a complete circle.

When we have a triangle, each of its angles is formed by the intersection of two of its sides. This means that the angles of a triangle are essentially measuring the amount of rotation needed to align one side of the triangle with another.

As you'll see in the video (above), if you take an object like a marker and rotate it across all three angles in a triangle, the marker will have flipped 180 degrees. You will see this by watching the yellow cover of the marker, which starts out on the left side of the marker and ends up on the right side of the marker. The marker rotates a full 180 degrees, equivalent to a straight line, after tracking the amount of rotation across all three angles in the triangle.

Make sure you watch the video to see this math right before your eyes!

It's not magic. It's math. And...it always adds up! :)

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