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Triangle Exterior Angle Theorem — Definition, Formula & Examples

The Triangle Exterior Angle Theorem states that an exterior angle of a triangle equals the sum of the two non-adjacent (remote) interior angles. This means the exterior angle is always larger than either of the remote interior angles individually.

Given triangle ABCABC with side BC‾\overline{BC} extended to point DD, the exterior angle ∠ACD\angle ACD satisfies m∠ACD=m∠A+m∠Bm\angle ACD = m\angle A + m\angle B, where ∠A\angle A and ∠B\angle B are the two interior angles of the triangle that are not adjacent to ∠ACD\angle ACD.

Key Formula

m∠ext=m∠1+m∠2m\angle_{\text{ext}} = m\angle_1 + m\angle_2
Where:
  • m∠extm\angle_{\text{ext}} = Measure of the exterior angle of the triangle
  • m∠1m\angle_1 = Measure of the first remote interior angle
  • m∠2m\angle_2 = Measure of the second remote interior angle

How It Works

When you extend one side of a triangle past a vertex, the angle formed between the extended side and the other side at that vertex is called an exterior angle. The two interior angles at the other two vertices are called the remote interior angles. To find the exterior angle, simply add those two remote interior angles together. This works because all three interior angles sum to 180°180°, and the exterior angle and its adjacent interior angle also sum to 180°180°, so the exterior angle must equal the sum of the remaining two.

Worked Example

Problem: In triangle PQR, angle P measures 50° and angle Q measures 70°. Side QR is extended to point S. Find the measure of exterior angle PRS.
Step 1: Identify the remote interior angles. The exterior angle is at vertex R, so the two remote interior angles are angle P and angle Q.
m∠P=50°,m∠Q=70°m\angle P = 50°, \quad m\angle Q = 70°
Step 2: Apply the Triangle Exterior Angle Theorem by adding the two remote interior angles.
m∠PRS=m∠P+m∠Q=50°+70°m\angle PRS = m\angle P + m\angle Q = 50° + 70°
Step 3: Calculate the result.
m∠PRS=120°m\angle PRS = 120°
Answer: The exterior angle PRS measures 120°.

Another Example

Problem: An exterior angle of a triangle measures 115°. One of the remote interior angles measures 40°. Find the other remote interior angle.
Step 1: Write the theorem with the known values. Let the unknown remote interior angle be x.
115°=40°+x115° = 40° + x
Step 2: Solve for x by subtracting 40° from both sides.
x=115°−40°=75°x = 115° - 40° = 75°
Answer: The other remote interior angle measures 75°.

Why It Matters

This theorem appears frequently in high school geometry proofs and on standardized tests like the SAT and ACT. Architects and engineers use it when calculating angles in triangular trusses and structural supports. It also serves as a stepping stone to more advanced results, such as the Exterior Angle Inequality Theorem used in college-level geometry courses.

Common Mistakes

Mistake: Using the adjacent interior angle instead of the two remote interior angles.
Correction: The adjacent interior angle is the one that shares the vertex with the exterior angle. It is NOT one of the angles you add. You need the two angles at the other vertices of the triangle.
Mistake: Confusing the exterior angle with a supplement and adding all three interior angles.
Correction: While the exterior angle is supplementary to its adjacent interior angle, the theorem directly gives you the exterior angle as the sum of only the two remote interior angles — not all three.

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