In a triangle, we can form exterior angles by extending the sides.

At a single vertex, we can create two different exterior angles by extending different sides.

These two angles may look different, but they are actually equal.

In this lesson, we will give a quick proof to show why these two exterior angles are always equal.

Rule

At any vertex of a triangle:

The two exterior angles are equal because each one forms a linear pair with the same interior angle.

Proof Idea

Consider a triangle with an interior angle Angle 3.

If we extend one side, we get an exterior angle Angle 4.

Angle 3 + Angle 4 = 180° (linear pair)

If we extend the other side at the same vertex, we get another exterior angle Angle 5.

Angle 3 + Angle 5 = 180° (linear pair)

So:

Angle 4 = 180° − Angle 3
Angle 5 = 180° − Angle 3

Therefore:

Angle 4 = Angle 5

So, the two exterior angles at a vertex are equal.

Example

If the interior angle at a vertex is 70°, find the two exterior angles.

Exterior angle = 180° − 70°

Exterior angle = 110°

So both exterior angles are 110°.

Video Explanation

Practice

• More Examples

• Take the Quiz

Continue Learning

1. Types of Triangles
2. Triangles (Basics)
3. Sum of Angles of a Triangle is 180° (Proof)
4. Find the Missing Angle of a Triangle
5. Sum of Exterior Angles of a Triangle (Proof)
6. Exterior Angle Theorem of a Triangle
7. Two Exterior Angles at a Vertex are Equal (Proof)
8. Two Exterior Angles are Equal at a Vertex (Quick Proof)

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