Two Exterior Angles are Equal at a Vertex (Quick Proof)

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.

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.

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 $110^\circ$.