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: Recall rhombus angle properties

Opposite angles of a rhombus are equal, and consecutive angles are supplementary. The given angle is $57^\circ$, so its consecutive angle is $180^\circ - 57^\circ = 123^\circ$.

Step2: Use diagonal angle bisector rule

Diagonals of a rhombus bisect the angles.

• $\angle 1$ bisects the $57^\circ$ angle: $m\angle 1 = \frac{57^\circ}{2} = 28.5^\circ$
• $\angle 4$ is equal to $\angle 1$ (alternate interior angles or symmetry): $m\angle 4 = 28.5^\circ$
• $\angle 2$ bisects the $123^\circ$ angle: $m\angle 2 = \frac{123^\circ}{2} = 61.5^\circ$
• $\angle 3$ is equal to $\angle 2$ (alternate interior angles or symmetry): $m\angle 3 = 61.5^\circ$

Answer:

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