GeoProof — Exterior Angle Theorem explorer

exterior angle = sum of the two remote interior angles

The Exterior Angle Theorem

Extend one side of a triangle past a vertex. The angle formed outside the triangle there — the exterior angle — equals the sum of the two interior angles at the other two vertices (the "remote" interior angles):

exterior angle = remote interior angle + remote interior angle

For example, the exterior angle at C equals angle A + angle B.

Why it's true

The interior angle and its exterior angle sit on a straight line, so they add to 180°. The three interior angles of a triangle also add to 180°. Subtracting the shared interior angle leaves exterior = the other two interior angles.

A consequence

An exterior angle is always larger than either remote interior angle on its own (since you're adding a positive amount). This fact — the "exterior angle inequality" — is used in many classical proofs.

Connection to polygons

Add up one exterior angle at each vertex, walking around any convex polygon, and the total is always 360° — the companion "sum of exterior angles" result.

A bit of history

The theorem is Proposition 32 in Book I of Euclid's Elements (c. 300 BCE), proved right alongside the fact that a triangle's angles sum to two right angles.