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Remote Interior Angles — Definition, Formula & Examples

Remote interior angles are the two interior angles of a triangle that are not next to (not adjacent to) a given exterior angle. They are sometimes called "non-adjacent interior angles."

Given a triangle with an exterior angle formed by extending one side, the remote interior angles are the two interior angles of the triangle that do not share a vertex with that exterior angle.

Key Formula

m∠exterior=m∠A+m∠Bm\angle{\text{exterior}} = m\angle{A} + m\angle{B}
Where:
  • m∠exteriorm\angle{\text{exterior}} = Measure of the exterior angle of the triangle
  • m∠Am\angle{A} = Measure of one remote interior angle
  • m∠Bm\angle{B} = Measure of the other remote interior angle

How It Works

When you extend one side of a triangle, the angle formed outside the triangle is an exterior angle. The interior angle at that same vertex is adjacent to the exterior angle, so the other two interior angles are the remote interior angles. By the Exterior Angle Theorem, the measure of the exterior angle equals the sum of the two remote interior angles. This relationship lets you find a missing angle without first calculating all three interior angles.

Worked Example

Problem: In triangle PQR, side QR is extended to point S, forming exterior angle PRS. If angle P = 50° and angle Q = 70°, find the measure of exterior angle PRS.
Identify the remote interior angles: The exterior angle PRS is at vertex R. The two angles that are not at vertex R are angle P and angle Q, so these are the remote interior angles.
Apply the Exterior Angle Theorem: The exterior angle equals the sum of the two remote interior angles.
m∠PRS=m∠P+m∠Q=50°+70°=120°m\angle PRS = m\angle P + m\angle Q = 50° + 70° = 120°
Answer: The exterior angle PRS measures 120°.

Why It Matters

SAT and ACT geometry questions frequently test the Exterior Angle Theorem, which relies on identifying remote interior angles. Recognizing them quickly saves time on multi-step problems involving triangle angle relationships and parallel-line proofs.

Common Mistakes

Mistake: Confusing the adjacent interior angle with a remote interior angle.
Correction: The adjacent interior angle shares a vertex and a side with the exterior angle — it is supplementary to it, not remote. The two remote interior angles are at the other two vertices of the triangle.

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