Exterior Angle of a Triangle
An exterior angle of a triangle is formed by one side and the extension of an adjacent side, and it equals the sum of the two non-adjacent interior angles.
Formula
\text{exterior angle} = \text{sum of two non-adjacent interior angles}
Definition
An exterior angle of a triangle is formed by one side and the extension of an adjacent side past a vertex. The Exterior Angle Theorem (Euclid I.32) states that it equals the sum of the two non-adjacent (remote) interior angles, and also equals $180^\circ$ minus the adjacent interior angle, since they form a linear pair: if $C$ is the interior angle and $D$ the exterior angle at $C$, then $A+B+C=180$ and $C+D=180$, so $D=A+B$.
Example
If a triangle has angles $50^\circ$, $60^\circ$, and $70^\circ$, extending the side past the $50^\circ$ vertex creates an exterior angle equal to $60+70=130^\circ$, its two "opposite" angles. For a triangle with $A=45$, $B=65$, $C=70$: the exterior angle at $C$ (extending side $BC$ past $C$) equals $A+B=110^\circ$, matching $180-70=110$ from the other method.
Key Insight
An exterior angle of a triangle is always bigger than either of the two opposite interior angles, the Exterior Angle Inequality (Euclid I.16), which is used to prove that the longest side of a triangle is opposite its largest angle, and that all exterior angles of a convex polygon sum to $360^\circ$. This inequality is more fundamental than the equality version: it can be proved without the parallel postulate, holding even in hyperbolic geometry, marking it as a theorem of "neutral" geometry, while the equality requires the parallel postulate, a distinction that reveals how deeply angle-sum relationships are tied to that single axiom.