Question

bus below, find the measures of $\angle 1$,
$m\angle 1$
$m\angle 2$
$m\angle 3$
$m\angle 4$

Explanation:

Step1: Identify rhombus angle properties

In a rhombus, opposite angles are equal, and adjacent angles are supplementary. Also, the diagonals bisect the angles. Given the labeled angle is $42^\circ$, $\angle 1 = 42^\circ$ (diagonal bisects the angle).

Step2: Find $\angle 2$

Adjacent angles in a rhombus sum to $180^\circ$. The angle supplementary to $2\times42^\circ$ is $180^\circ - 84^\circ = 96^\circ$. The diagonal bisects this angle, so $\angle 2 = \frac{96^\circ}{2} = 48^\circ$.

Step3: Find $\angle 3$

$\angle 3$ is equal to $\angle 1$ (opposite angles bisected equally, or alternate interior angles in parallel sides of rhombus), so $\angle 3 = 42^\circ$.

Step4: Find $\angle 4$

$\angle 4$ is equal to $\angle 2$ (opposite angles bisected equally, or alternate interior angles in parallel sides of rhombus), so $\angle 4 = 48^\circ$.

Answer:

$m\angle 1 = 42^\circ$
$m\angle 2 = 48^\circ$
$m\angle 3 = 42^\circ$
$m\angle 4 = 48^\circ$