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 \(\text{Angle 3}\).
If we extend one side, we get an exterior angle \(\text{Angle 4}\).
$$\text{Angle 3 + Angle 4 }= 180^\circ \text{ (linear pair)}$$
If we extend the other side at the same vertex, we get another exterior angle \(\text{Angle 5}\).
$$\text{Angle 3 + Angle 5 } = 180^\circ \text{ (linear pair)}$$
So:
$$\text{Angle 4 } = 180^\circ − \text{Angle 3}$$
$$\text{Angle 5 } = 180^\circ − \text{Angle 3}$$
Therefore:
$$\text{Angle 4 } = \text{Angle 5}$$
So, the two exterior angles at a vertex are equal.
Example
If the interior angle at a vertex is \(70^\circ\), find the two exterior angles.
$$\text{Exterior angle } = 180^\circ − 70^\circ$$
$$\text{Exterior angle } = 110^\circ$$
So both exterior angles are \(110^\circ\).
Video Explanation
Practice
Continue Learning
- Sum of Exterior Angles of a Triangle (Proof)
- Exterior Angle Theorem of a Triangle
- Two Exterior Angles at a Vertex are Equal (Proof)
- Two Exterior Angles are Equal at a Vertex (Quick Proof) (Current lesson)