Answer
$m \angle 1 = 32^{\circ}$
$m \angle 2 = 90^{\circ}$
$m \angle 3 = 58^{\circ}$
$m \angle 4 = 32^{\circ}$
Work Step by Step
Diagonals of rhombuses bisect pairs of opposite angles, so $m \angle 3$ is also $58^{\circ}$.
Diagonals of rhombuses cross each other at right angles, so $m \angle 2 = 90^{\circ}$.
Let's look at one of the smaller triangles. We have $m \angle 2$, the angle that is bisected by the diagonal that also gave us $\angle 3$; therefore, we only have to find $m \angle 1$.
The interior angles of a triangle add up to $180^{\circ}$, so let's set up an equation where we can find the measure of one of the interior angles given the measures of the other two angles:
$m \angle 1 = 180 - (90 + 58)$
Evaluate parentheses first, according to order of operations:
$m \angle 1 = 180 - (148)$
Subtract to solve:
$m \angle 1 = 32^{\circ}$
Now that we know $m \angle 1$, we also know $m \angle 4$ because the diagonal bisected the angle containing both $\angle 1$ amd $\angle 4$, so $m \angle 4$ is the same as $m \angle 1$.
$m \angle 4 = 32^{\circ}$
