Question

for the rhombus below, find the measures of ∠1, ∠2, ∠3, and ∠4.\
your answer is incorrect.\

• ( mangle 1 ): your answer is incorrect.\
• ( mangle 3 ): your answer is incorrect.\
• ( mangle 4 ): your answer is incorrect.

Explanation:

Step1: Recall rhombus side/angle properties

All sides of a rhombus are equal; diagonals bisect vertex angles, and adjacent angles are supplementary. Also, diagonals are perpendicular, forming right triangles.

Step2: Solve for $m\angle1$

$\angle1$ is congruent to the given $31^\circ$ angle (diagonal bisects the vertex angle, and opposite angles of a rhombus are equal).
$m\angle1 = 31^\circ$

Step3: Solve for $m\angle2$

In the right triangle formed by diagonals, angles sum to $180^\circ$.
$m\angle2 = 90^\circ - 31^\circ = 59^\circ$

Step4: Solve for $m\angle3$

$\angle3$ is congruent to $\angle2$ (diagonal bisects the adjacent vertex angle; adjacent angles of rhombus are supplementary, so the bisected angle matches $\angle2$).
$m\angle3 = 59^\circ$

Step5: Solve for $m\angle4$

In the right triangle formed by diagonals, angles sum to $180^\circ$.
$m\angle4 = 90^\circ - 59^\circ = 31^\circ$

Answer:

$m\angle1 = 31^\circ$
$m\angle2 = 59^\circ$
$m\angle3 = 59^\circ$
$m\angle4 = 31^\circ$