Answer
$m \angle 1= 105^{\circ}$
$m \angle 2 = 75^{\circ}$
$m \angle 3 = 75^{\circ}$
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 $105^{\circ}$, the other base angle, $\angle 1$, is $105^{\circ}$.
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, $\angle 2$ and $\angle 3$:
$m \angle 2 = 180 - 105$
Subtract to solve:
$m \angle 2 = 75^{\circ}$
If $m \angle 2 = 75^{\circ}$, then $m \angle 3 = 75^{\circ}$ because these two angles are base angles of the isosceles trapezoid and, thus, are congruent.
