Answer
$m \angle 1 = 90^{\circ}$
$m \angle 3 = 90^{\circ}$
$m \angle 2 = 26^{\circ}$
Work Step by Step
According to theorem 6-22, the diagonals of a kite form perpendicular angles.
Therefore, $m \angle 1 = m \angle 3 = 90^{\circ}$.
For one of the smaller angles, we have one angle measuring $90^{\circ}$ and another angle measuring $64^{\circ}$. 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 third angle = $180 - (90 + 64)$
Evaluate parentheses first:
m third angle = $180 - (154)$
Subtract to solve:
m third angle = $26^{\circ}$
We can see that we have two triangles that are congruent by SSS. Corresponding parts of congruent angles are congruent. Therefore, $\angle 2$ is congruent to the third angle we just found:
$m \angle 2 = 26^{\circ}$
