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: $Δ X Y Z$

Prove: $m ∠ 1 + m ∠ 2 + m ∠ 3 = 180 °$

Math diagram

Statements	Reasons
Construct a line parallel to $\bar{X Z}$ through point $Y$ . Call this line $\overset{\leftrightarrow}{A Y}$	Construction
$m ∠ 1 + m ∠ 5 = m ∠ A Y X$	Angle Addition Postulate
$m ∠ A Y X + m ∠ 4 = 180 °$	Linear Pair Postulate
$m ∠ 1 + m ∠ 5 + m ∠ 4 = 180 °$	Substitution
$\begin{array}{l} ∠ 2 ≅ ∠ 4 \\ ∠ 3 ≅ ∠ 5 \end{array}$	Alternate Interior Angles Theorem
$\begin{array}{l} m ∠ 2 = m ∠ 4 \\ m ∠ 3 = m ∠ 5 \end{array}$	Definition of Congruence
$m ∠ 1 + m ∠ 3 + m ∠ 2 = 180 °$	Substitution