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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: m∠1+m∠2+m∠3=180°

Statements Reasons

Construct a line parallel to XZ ¯ through point Y . Call this line AY ↔

Construction
m∠1+m∠5=m∠AYX Angle Addition Postulate
m∠AYX+m∠4=180° Linear Pair Postulate
m∠1+m∠5+m∠4=180° Substitution
∠2≅∠4 ∠3≅∠5 Alternate Interior Angles Theorem
m∠2=m∠4 m∠3=m∠5

Definition of Congruence
m∠1+m∠3+m∠2=180° Substitution