Question

for the rhombus below, find the measures of $\angle 1$, $\angle 2$, $\angle 3$, and $\angle 4$.
$m\angle 1 = \square^\circ$
$m\angle 2 = \square^\circ$
$m\angle 3 = \square^\circ$
$m\angle 4 = \square^\circ$

Explanation:

Step1: Identify rhombus angle properties

In a rhombus, opposite angles are equal, and adjacent angles are supplementary. Also, the diagonal bisects the angles.

Step2: Find $m\angle2$

The diagonal bisects the vertex angle, so $m\angle2 = 32^\circ$.

Step3: Calculate $m\angle1$

Adjacent angles in a rhombus are supplementary. The sum of angles in a triangle is $180^\circ$, so:
$$m\angle1 = 180^\circ - 32^\circ - 32^\circ = 116^\circ$$

Step4: Find $m\angle3$

Opposite angles in a rhombus are equal, so $m\angle3 = m\angle1 = 116^\circ$.

Step5: Find $m\angle4$

The diagonal bisects the opposite vertex angle, so $m\angle4 = m\angle2 = 32^\circ$.

Answer:

$m\angle1 = 116^\circ$
$m\angle2 = 32^\circ$
$m\angle3 = 116^\circ$
$m\angle4 = 32^\circ$