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
The measure of the exterior angle of a triangle is equal to the sum of the measure sf its interior angles.
$latex m \angle 3 + m \angle 4 = 180$ 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, $latex 180 – m \angle 3$,
$latex m \angle 1 + m \angle 2 = m \angle 4. \blacksquare$