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:
Start with the following triangle with arbitrary values for the angles:
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

Jul 06, 18 12:29 PM
Learn how to solve two types of logarithmic equations
Read More
New math lessons
Your email is safe with us. We will only use it to inform you about new math lessons.