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: A triangle exterior angle equals the sum of the two non-adjacent interior angles.

Common stuck point: The procedure for exterior angle theorem is the easy part; the trap is using the adjacent interior angle as a remote angle. Asking "Which two interior angles are not touching the exterior angle?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Which two interior angles are not touching the exterior angle?

Worked Examples

Example 1

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

Answer

The exterior angle is 113°113°.

First step

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.

Full solution

  1. 2
    Step 2: The two remote interior angles are 65°65° and 48°48°.
  2. 3
    Step 3: Exterior angle =65°+48°=113°= 65° + 48° = 113°.
The Exterior Angle Theorem provides a shortcut: instead of finding the third interior angle first (180°−65°−48°=67°180° - 65° - 48° = 67°) and then its supplement (180°−67°=113°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 △ABC\triangle ABC, the exterior angle at CC is 130°130°. If ∠A=5x°\angle A = 5x° and ∠B=3x+2°\angle B = 3x + 2°, find the value of xx and both interior angles.

Example 3

medium
In △ABC\triangle ABC, the exterior angle at CC measures 4x+10°4x + 10°. The remote interior angles measure 2x+5°2x + 5° and x+25°x + 25°. Find xx.

Example 4

medium
In △ABC\triangle ABC, ∠A=38°\angle A = 38° and ∠B=62°\angle B = 62°. Find the exterior angle at CC and the exterior angle at AA.

Example 5

medium
Two remote interior angles of a triangle differ by 20°20° and the exterior angle is 110°110°. Find both remote interior angles.

Example 6

medium
An exterior angle of a triangle is twice the smallest remote interior angle. If the exterior angle is 80°80°, find both remote interior angles.

Example 7

medium
In △ABC\triangle ABC, the exterior angle at BB is 145°145° and the interior angle at CC is 50°50°. Find ∠A\angle A and ∠B\angle B.

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°140°. One of the remote interior angles is 75°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.

Example 3

easy
Two remote interior angles of a triangle measure 52°52° and 68°68°. Find the exterior angle at the third vertex.

Example 4

easy
An exterior angle of a triangle measures 150°150°. What is the adjacent interior angle?

Example 5

medium
The exterior angle of a triangle is 125°125° and the two remote interior angles are in the ratio 2:32:3. Find each remote interior angle.

Example 6

medium
In △ABC\triangle ABC, the exterior angle at AA is 130°130°. If ∠B=3∠C\angle B = 3 \angle C, find ∠B\angle B and ∠C\angle C.

Example 7

easy
The exterior angle of a triangle at vertex CC is 75°75°. One remote interior angle is 35°35°. Find ∠C\angle C (the interior angle at CC).

Example 8

medium
In △ABC\triangle ABC, side BCBC is extended to DD. If ∠ACD=132°\angle ACD = 132° and ∠B=64°\angle B = 64°, find ∠A\angle A.

Example 9

hard
In △PQR\triangle PQR, the exterior angle at RR is (5x−10)°(5x - 10)° and the remote interior angles are (2x+15)°(2x + 15)° and (2x−5)°(2x - 5)°. Find xx and all three interior angles.

Example 10

medium
An exterior angle of an isosceles triangle, formed at the vertex (apex), is 140°140°. Find each base angle.

Example 11

easy
The exterior angle at one vertex of an equilateral triangle is what?

Example 12

hard
Prove using the angle-sum theorem that every exterior angle of a triangle equals the sum of the two non-adjacent interior angles.

Example 13

medium
In △ABC\triangle ABC, ∠A=3y\angle A = 3y, ∠B=2y\angle B = 2y, and the exterior angle at CC is 150°150°. Find yy and ∠C\angle C.

Example 14

hard
In △ABC\triangle ABC, the exterior angles at AA, BB, and CC are in the ratio 5:6:75:6:7. Find each interior angle.

Example 15

easy
What is the sum of the three exterior angles of any triangle (one at each vertex)?

Example 16

medium
Two exterior angles of a triangle are 110°110° and 130°130°. Find the third exterior angle and all three interior angles.

Example 17

challenge
In △ABC\triangle ABC, point DD lies on the extension of BCBC beyond CC. The bisector of ∠ACD\angle ACD meets the extension of BABA at EE. If ∠B=40°\angle B = 40° and ∠A=60°\angle A = 60°, find ∠AEC\angle AEC.

Example 18

hard
An exterior angle of a triangle is 120°120°. Suppose this same triangle's interior angles are also in arithmetic progression. Find all three interior angles.
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Background Knowledge

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

triangle angle sumangles