Exterior Angle Theorem

Geometry · Axiom Academy

Why an exterior angle of a triangle is exactly the sum of the two interior angles it does not touch. When you extend side AC past vertex C, the new angle outside the triangle is the exterior angle ∠4 . The two interior angles it does not touch — ∠1 at A and ∠2 at B — are its remote interior angles . The exterior angle equals the sum of the two remote interior angles Two facts you already know combine to force the result. First, the three interior angles of any triangle add to 180° . Second, the interior angle ∠3 at C and the exterior angle ∠4 form a straight line — a linear pair , so they are supplementary. ∠3 (interior) and ∠4 (exterior) sit on one straight line, so ∠3 + ∠4 = 180°. ∠3 appears on both sides of the equation and drops out, leaving ∠4 = ∠1 + ∠2. In triangle ABC, ∠A = 45° and ∠B = 70° . Find the measure of the exterior angle at C. The exterior angle at C has remote interior angles at A and B: ∠A = 45° and ∠B = 70°. Check it against the interior angle The interior angle at C is 180° − (45° + 70°) = 65°. Its supplement is 180° − 65° = 115° — the same exterior angle, exactly as the theorem predicts. You've seen what an exterior angle is, proven the theorem from the triangle's angle sum, and used it to find an exterior angle of 115°. Scroll up to revisit any step.

This is the written version of the interactive lesson above. See the full Geometry course.