Exterior Angle Theorem

A proof that an exterior angle of a triangle equals the sum of the two non-adjacent interior angles.

Keywords: exterior angle, interior angle, angle sum, supplementary angles

Prerequisites: angle-sum-proof · Difficulty: beginner

An exterior angle of a triangle equals the sum of the two non-adjacent interior angles:

Strategy: extend side $BC$ past $C$ to point $D$. The exterior angle $\angle ACD$ and the interior angle $\angle ACB$ are supplementary, summing to $180°$. Combined with the angle sum $\angle A+\angle B+\angle ACB=180°$, subtracting $\angle ACB$ gives $\angle ACD=\angle A+\angle B$.

Step 1: Extend a Side

Start with triangle $ABC$ and extend side $BC$ past $C$ to point $D$. This creates the exterior angle $\angle ACD$.

Step 2: Supplementary Angles

The exterior angle $\angle ACD$ and the interior angle $\angle ACB$ are supplementary — they form a straight line and sum to $180°$:

Step 3: Exterior Angle $=\angle A+\angle B$

By the angle sum theorem: $\angle A+\angle B+\angle ACB=180°$.

Since $\angle ACD+\angle ACB=180°$ as well, subtracting $\angle ACB$ from both:

The exterior angle equals the sum of the two remote interior angles!

The exterior angle of a triangle equals the sum of the two remote interior angles:

This is often more convenient than the full angle sum theorem for calculating unknown angles. ∎