Question

incorrect
your answer is incorrect.

• ( mangle 1 ): your answer is incorrect.
• ( mangle 2 ): your answer is incorrect.
• ( mangle 3 ): your answer is incorrect.
• ( mangle 4 ): your answer is incorrect.

for the rhombus below, find the measures of ( angle 1 ), ( angle 2 ), ( angle 3 ), and ( angle 4 ).
( mangle 1 = 28.5^circ )
( mangle 2 = 61.5^circ )
( mangle 3 = 61.5^circ )
( mangle 4 = 28.5^circ )

Explanation:

Step1: Recall rhombus angle properties

All sides of a rhombus are equal, so its opposite angles are equal, and adjacent angles are supplementary. Also, the diagonals of a rhombus bisect the vertex angles.

Step2: Find adjacent angle to 57°

Adjacent angles sum to $180^\circ$.
$180^\circ - 57^\circ = 123^\circ$

Step3: Calculate $m\angle1$ and $m\angle4$

Diagonal bisects the $57^\circ$ angle.
$m\angle1 = m\angle4 = \frac{57^\circ}{2} = 28.5^\circ$

Step4: Calculate $m\angle2$ and $m\angle3$

Diagonal bisects the $123^\circ$ angle.
$m\angle2 = m\angle3 = \frac{123^\circ}{2} = 61.5^\circ$

Wait, correction: The labeled $57^\circ$ angle is formed by a side and a diagonal, not a vertex angle. Let's redo:

Step1: Use rhombus diagonal right angle

Diagonals of a rhombus are perpendicular, so the triangle with $57^\circ$ is right-angled.

Step2: Solve for $m\angle2$

In right triangle, angles sum to $180^\circ$.
$m\angle2 = 90^\circ - 57^\circ = 33^\circ$

Step3: $\angle3$ equals $\angle2$

Diagonal bisects the vertex angle.
$m\angle3 = m\angle2 = 33^\circ$

Step4: Solve for $m\angle1$

In the top triangle, use angle sum.
$m\angle1 = 180^\circ - 90^\circ - 33^\circ = 57^\circ$

Step5: $\angle4$ equals $\angle1$

Diagonal bisects the opposite vertex angle.
$m\angle4 = m\angle1 = 57^\circ$

Answer:

$m\angle1 = 57^\circ$, $m\angle2 = 33^\circ$, $m\angle3 = 33^\circ$, $m\angle4 = 57^\circ$