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Exterior Angle Theorem

The Exterior Angle Theorem states that an exterior angle of a triangle is equal to the sum of the two interior angles that are not next to it (the non-adjacent, or remote, interior angles).

In any triangle, an exterior angle formed by extending one side of the triangle is equal in measure to the sum of the two non-adjacent interior angles. If a triangle has interior angles AA, BB, and CC, and the exterior angle is adjacent to angle CC, then the exterior angle equals A+BA + B. This theorem follows directly from the fact that the three interior angles of a triangle sum to 180°180° and that an exterior angle and its adjacent interior angle are supplementary.

Key Formula

d=A+Bd = A + B
Where:
  • dd = the exterior angle of the triangle
  • AA = one of the two non-adjacent (remote) interior angles
  • BB = the other non-adjacent (remote) interior angle

Worked Example

Problem: In triangle PQR, angle P = 50° and angle Q = 65°. Side QR is extended beyond R to a point S, forming exterior angle QRS. Find the measure of the exterior angle at R.
Step 1: Identify the two non-adjacent interior angles. The exterior angle is at vertex R, so the non-adjacent interior angles are P and Q.
Step 2: Apply the Exterior Angle Theorem: the exterior angle equals the sum of the two remote interior angles.
d=∠P+∠Qd = \angle P + \angle Q
Step 3: Substitute the known values and compute.
d=50°+65°=115°d = 50° + 65° = 115°
Step 4: You can verify this by first finding the interior angle at R, then checking that it and the exterior angle are supplementary.
∠R=180°−50°−65°=65°,65°+115°=180°  ✓\angle R = 180° - 50° - 65° = 65°, \quad 65° + 115° = 180° \; \checkmark
Answer: The exterior angle at R measures 115°.

Visualization

Why It Matters

The Exterior Angle Theorem gives you a shortcut for finding unknown angles in triangles without needing to calculate all three interior angles first. It appears frequently in geometry proofs, standardized tests, and any situation involving angle relationships—such as designing roof trusses, analyzing structural supports, or solving multi-triangle problems where exterior angles connect one triangle's measurements to another.

Common Mistakes

Mistake: Using the adjacent interior angle instead of the two non-adjacent ones.
Correction: The theorem only involves the two remote (non-adjacent) interior angles. The adjacent interior angle is supplementary to the exterior angle, not equal to it.
Mistake: Confusing the exterior angle with the interior angle at the same vertex.
Correction: An exterior angle is formed by one side of the triangle and the extension of the other side at that vertex. It and the interior angle at the same vertex always add up to 180°.

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