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

The exterior angle theorem states that an exterior angle of a triangle equals the sum of the two non-adjacent interior angles:

\angle ACD = \angle A + \angle B
    

This is a direct consequence of the triangle angle sum theorem.

Proof Strategy

Extend side $BC$ past $C$ to point $D$. The exterior angle $\angle ACD$ and the interior angle $\angle ACB$ are supplementary (sum to $180°$). By the angle sum theorem, $\angle A + \angle B + \angle ACB = 180°$. Subtracting $\angle ACB$ from both equations gives $\angle ACD = \angle A + \angle B$.

The Proof

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 $180°$:

\angle ACD + \angle ACB = 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:

\angle ACD = \angle A + \angle B
    

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

Conclusion

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

\angle ACD = \angle A + \angle B
    

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