Answer
$m \angle 1 = 90^{\circ}$
$m \angle 2 = 55^{\circ}$
$m \angle 3 = 90^{\circ}$
$m \angle 4 = 55^{\circ}$
$m \angle 5 = 35^{\circ}$
Work Step by Step
According to theorem 6-22, the diagonals of a kite form perpendicular angles.
Therefore, $m \angle1 = m \angle 3 = 90^{\circ}$.
For the triangle containing $\angle 1$ and $\angle 2$, we have the measure of two of the angles already. We have $m \angle 1 = 90^{\circ}$ and another angle measuring $35^{\circ}$. 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, $\angle 2$:
$m \angle 2 = 180 - (90 + 35)$
Evaluate parentheses first:
$m \angle 2 = 180 - (125)$
Subtract to solve:
$m \angle 2 = 55^{\circ}$
We have two triangles that are congruent by SSS. Corresponding parts of congruent angles are congruent. Therefore:
$m \angle 4 ≅ m \angle 2 = 55^{\circ}$
$m \angle 5 = 35^{\circ}$
