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 XZ ¯ through point Y . Call this line AY

m1+m5=mAYX Angle Addition Postulate
mAYX+m4=180° Linear Pair Postulate
m1+m5+m4=180° Substitution
24 35 Alternate Interior Angles Theorem
m2=m4 m3=m5

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