# Proof of the angle sum theorem

Angle sum theorem: The angle measures in any triangles add up to 180 degrees.

Key concept: Alternate interior angles are equal. We will accept this fact without a proof. The figure above shows two pairs of alternate interior angles.

For the pair in red, angle 1 = angle 2. For the pair in blue, angle 3 = angle 4

Now, take a close look at the figure below. I claim that angle x is equal to 85 degrees so the sum is 180 degrees. To see why this is so, draw a line parallel to AC at vertex B Angle a = 65 degrees because it alternates with the angle inside the triangle that measures 65 degrees

Angle b = 30 degrees because it alternates with the angle inside the triangle that measures 30 degrees

Looking at the figure again, it is easy to see why angle x is 85.

Since the 65 degrees angle, the angle x, and the 30 degrees angle make a straight line together, the sum must be 180 degrees

Since, 65 + angle x + 30 = 180, angle x must be 85

This is not a proof yet. This just shows that it works for one specific example

Proof of the angle sum theorem: Since angle a, angle b, and angle c make a straight line,

angle a + angle b + angle c = 180 degrees

Since alternate interior angles are equal, angle a = angle x and angle b = angle y

Therefore, angle x + angle y + angle c = 180 degrees

## Recent Articles 1. ### How to Write an Inequality

Dec 12, 19 07:51 AM

Learn how to write an inequality quickly with this easy to follow math lesson.

New math lessons

Your email is safe with us. We will only use it to inform you about new math lessons.

Tough Algebra Word Problems.

If you can solve these problems with no help, you must be a genius! Real Life Math Skills

Learn about investing money, budgeting your money, paying taxes, mortgage loans, and even the math involved in playing baseball.