the exterior-angle theorem

Extend one side of a triangle past a vertex and you open up an angle outside the triangle, between the extended side and the next side. That is an exterior angle. The exterior-angle theorem makes a clean claim about it: an exterior angle of a triangle equals the sum of the two interior angles that do not touch it — the two so-called remote interior angles at the other two vertices.

Precisely: in triangle ABC, extend side BC beyond C to a point D. The exterior angle ACD then satisfies m(angle ACD) = m(angle A) + m(angle B). The proof falls right out of the angle-sum theorem: angle ACD and the interior angle ACB form a linear pair, so m(angle ACD) = 180 - m(angle ACB); but the angle sum gives m(angle A) + m(angle B) = 180 - m(angle ACB); the two right-hand sides match, so the exterior angle equals the sum of the two remote interior angles. A handy corollary: an exterior angle is strictly larger than either remote interior angle alone.

This theorem is a constant tool in angle chasing and in proofs about the largest angle and longest side. Watch one trap: the theorem pairs an exterior angle with the two remote (far) interior angles, not the adjacent one. The adjacent interior angle is the exterior angle's supplement, while the remote pair is its equal — different relationships, easily confused.