Question

for the rhombus below, find the measures of $\angle 1$, $\angle 2$, $\angle 3$, and $\angle 4$.

Explanation:

Step1: Use rhombus angle properties

In a rhombus, opposite angles are equal, and adjacent angles are supplementary. The given angle is $57^\circ$, so its adjacent angle is $180^\circ - 57^\circ = 123^\circ$. The diagonals of a rhombus bisect the angles, so $\angle 1 = \frac{57^\circ}{2} = 28.5^\circ$.

Step2: Find $\angle 2$ using bisection

$\angle 2$ is half of the adjacent angle to $57^\circ$, so $\angle 2 = \frac{180^\circ - 57^\circ}{2} = \frac{123^\circ}{2} = 61.5^\circ$.

Step3: Identify $\angle 3$ via rhombus sides

All sides of a rhombus are equal, so the triangle containing $\angle 1$ and $\angle 3$ is isosceles. $\angle 3$ equals $\angle 1$, so $\angle 3 = 28.5^\circ$.

Step4: Identify $\angle 4$ via symmetry

By the symmetry of the rhombus, $\angle 4$ equals $\angle 2$, so $\angle 4 = 61.5^\circ$.

Answer:

$m\angle 1 = 28.5^\circ$
$m\angle 2 = 61.5^\circ$
$m\angle 3 = 28.5^\circ$
$m\angle 4 = 61.5^\circ$