Answer
$m \angle 1 = 60^{\circ}$
$m \angle 2 = 90^{\circ}$
$m \angle 3 = 30^{\circ}$
Work Step by Step
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 one angle measuring $90^{\circ}$ and an angle that measures $60^{\circ}$; therefore, we only have to find the angle adjacent to $\angle 3$.
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:
$m$ third angle = $180 - (90 + 60)$
Evaluate parentheses first, according to order of operations:
$m$ third angle = $180 - (150)$
Subtract to solve:
$m$ third angle = $30$
Now that we know the angle adjacent to $\angle 3$, we also know $m \angle 3$ because the two angles are part of the larger angle that is bisected by a diagonal, so these angles are congruent:
$m \angle 3 = 30^{\circ}$
For $m \angle 1$, it is also $60^{\circ}$ because it is adjacent to the angle marked $60^{\circ}$ and is the resulting angle when that bigger angle is bisected by the diagonal.
$m \angle 1 = 60^{\circ}$
