Angle Sum Theorem

The Angle Sum Theorem gives an important result about triangles, which is used in many algebra and geometry problems. We give the proof below.

Theorem: The sum of the measures of the interior angles of a triangle is 180°.

Given: ΔXYZ

Prove: m1 + m2 + m3 = 180°

Statements Reasons

Construct a line parallel to through point Y. Call this line

m1 + m5 = mAYX Angle Addition Postulate
mAYX + m4 = 180° Linear Pair Postulate
m1 + m5 + m4 = 180° Substitution
Alternate Interior Angles Theorem

m2 = m4

m3 = m5

Definition of Congruence
m1 + m3 + m2 = 180° Substitution