# The Exterior Angle Theorem

Introduction

If one side of a triangle is extended, an exterior angle is formed.  An exterior angle of a triangle forms a linear pair with the adjacent interior angle. The two non-adjacent interior angles to the exterior angle are called its remote interior angles. In the figure below, \$latex \angle 1\$ and \$latex \angle 2\$ are remote angles of \$latex \angle 4\$. In the second triangle above, we can see that the sum of the measures of the remote angles is equal to the measure of the exterior angle. That is, \$latex \angle 1 + \angle 2 = \angle 4\$. Is this observation always true? In this post, we prove that it is indeed true

Theorem

The measure of the exterior angle of a triangle is equal to the sum of the measure sf its interior angles.

Proof

\$latex m \angle 3 + m \angle 4 = 180\$ since the two angles are linear pair and therefore supplementary.

\$latex m \angle 4 = 180 – m \angle 3\$* by Addition Property of Equality

\$latex m \angle 1 + m \angle 2 + m \angle 3 = 180\$  because the angle sum of a triangle is \$latex 180\$ degrees.

\$latex m \angle 1 + m \angle 2 = 180 – m \angle 3\$ by Addition Property of Equality**

From * and **,  \$latex m \angle 1 + m \angle 2\$ and \$latex m \angle 4\$ both equal, \$latex 180 – m \angle 3\$,

\$latex m \angle 1 + m \angle 2 = m \angle 4. \blacksquare\$

## 3 thoughts on “The Exterior Angle Theorem”

1. Zack Maddox on said:

This should be proven in neutral geometry. You cannot assume triangles have angle sum 180 in neutral geometry. That’s Euclidean geometry only.