Exterior Angle Theorem Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Exterior Angle Theorem.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

An exterior angle of a triangle equals the sum of the two non-adjacent (remote) interior angles.

Imagine standing at one corner of a triangular park and looking along one side. The exterior angle is how far you'd turn to look back along the other side. That turn combines the 'bends' at the other two corners—it equals their angles added together.

Read the full concept explanation →

How to Use These Examples

  • Read the first worked example with the solution open so the structure is clear.
  • Try the practice problems before revealing each solution.
  • Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: An exterior angle captures the combined turning from the two far corners of the triangle.

Common stuck point: The exterior angle is supplementary to its adjacent interior angle (\text{exterior} + \text{adjacent interior} = 180°), which is how this theorem follows from the angle sum property.

Worked Examples

Example 1

easy
In a triangle, two interior angles are 65° and 48°. An exterior angle is formed at the third vertex. Find the exterior angle.

Solution

  1. 1
    Step 1: The Exterior Angle Theorem states: an exterior angle of a triangle equals the sum of the two non-adjacent (remote) interior angles.
  2. 2
    Step 2: The two remote interior angles are 65° and 48°.
  3. 3
    Step 3: Exterior angle = 65° + 48° = 113°.

Answer

The exterior angle is 113°.
The Exterior Angle Theorem provides a shortcut: instead of finding the third interior angle first (180° - 65° - 48° = 67°) and then its supplement (180° - 67° = 113°), you directly add the two remote interior angles. Both methods give the same answer because they both reduce to the same algebra.

Example 2

medium
In \triangle ABC, the exterior angle at C is 130°. If \angle A = 5x° and \angle B = 3x + 2°, find the value of x and both interior angles.

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy
An exterior angle of a triangle measures 140°. One of the remote interior angles is 75°. Find the other remote interior angle.

Example 2

hard
Prove that an exterior angle of a triangle is always greater than either of the two remote interior angles.
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Background Knowledge

These ideas may be useful before you work through the harder examples.

triangle angle sumangles