Answer
$m \angle 1 = 90^{\circ}$
$m \angle 2 = 45^{\circ}$
$m \angle 3 = 45^{\circ}$
Work Step by Step
According to theorem 6-22, the diagonals of a kite form perpendicular angles.
Therefore, $m \angle 1 = 90^{\circ}$.
For the triangle containing $\angle 1$, we have the measure of two of the angles already. We have one angle measuring $90^{\circ}$ and another angle measuring $45^{\circ}$. Using the triangle sum theorem, we can subtract the sum of the two known angles from $180^{\circ}$ and get the measure of the third angle, $\angle 2$:
$m \angle 2 = 180 - (90 + 45)$
Evaluate parentheses first:
$m \angle 2 = 180 - (135)$
Subtract to solve:
$m \angle 2 = 45^{\circ}$
We can see that we have two triangles that are congruent by SSS. Corresponding parts of congruent angles are congruent. Therefore:
$m \angle 3 = 45^{\circ}$
