The Exterior Angle Property (also called the Exterior Angle Theorem) is one of the most important properties of a triangle in geometry. It is a direct consequence of the angle sum property of triangles.
The Exterior Angle Property also proves a useful corollary: An exterior angle of a triangle is always greater than either of the non-adjacent interior angles. This is because it equals their sum, and both angles are positive.
Statement: The exterior angle of a triangle is equal to the sum of the two non-adjacent (opposite) interior angles.
If a triangle has interior angles ∠A, ∠B, and ∠C, and an exterior angle ∠ACD is formed by extending the side BC:
∠ACD = ∠A + ∠B
Proof (brief): We know that the sum of all interior angles of a triangle = 180° → ∠A + ∠B + ∠ACB = 180° ...(1)
Also, ∠ACB and ∠ACD are supplementary (they form a straight line, so they add up to 180°) → ∠ACB + ∠ACD = 180° ...(2)
From (1) and (2): ∠ACD = ∠A + ∠B ✓
Q1. In a triangle, two interior angles are 65° and 45°. Find the exterior angle at the third vertex. Solution: Exterior angle = 65° + 45° = 110° ✓
Q2. An exterior angle of a triangle is 110°. One of the non-adjacent interior angles is 60°. Find the other. Solution: 110° = 60° + x → x = 110° − 60° = 50° ✓
Q3. An exterior angle of a triangle is 3 times one of the non-adjacent angles. If that angle is 40°, find the exterior angle and the other non-adjacent angle. Solution: Exterior angle = 3 × 40° = 120°. Other angle = 120° − 40° = 80° ✓
Only if one of the non-adjacent interior angles is 0°, which is impossible in a real triangle. So in practice, no — an exterior angle is always strictly greater than each of the non-adjacent interior angles.
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