Answer
$m \angle 3 = 111^{\circ}$
$m \angle 2 = 69^{\circ}$
$m \angle 1 = 69^{\circ}$
Work Step by Step
According to theorem 6-19, the base angles of an isosceles trapezoid are congruent; therefore, if one of the base angles is $111^{\circ}$, the other base angle, $\angle 3$, is $111^{\circ}$.
If we take a look at the diagram, we see that we essentially have two transversals cutting a pair of parallel lines. The angles formed by each transversal are, in actuality, alternate interior angles; these types of angles are supplementary.
Let's set up the equation to find the other two base angles, $\angle 1$ and $\angle 2$:
$m \angle 2 = 180 - 111$
Subtract to solve:
$m \angle 2 = 69^{\circ}$
If $m \angle 2 = 69^{\circ}$, then $m \angle 1 = 69^{\circ}$ because these two angles are base angles of the isosceles trapezoid and, thus, are congruent.
