Question

quadrilateral pqrs is a rhombus and ( mangle prs = 2x ). what is the value of ( x )?

Explanation:

Step1: Use rhombus parallel sides property

In rhombus PQRS, $QR \parallel PS$, so $\angle PQR$ and $\angle QPS$ are supplementary:
$m\angle QPS = 180^\circ - 56^\circ = 124^\circ$

Step2: Use rhombus diagonal angle bisector property

Diagonal PR bisects $\angle QPS$, so $m\angle RPS = \frac{1}{2}m\angle QPS$
$m\angle RPS = \frac{1}{2} \times 124^\circ = 62^\circ$

Step3: Use alternate interior angles theorem

Since $QR \parallel PS$, $\angle PRS = \angle RPS$ (alternate interior angles). Set equal to $2x$:
$2x = 62^\circ$

Step4: Solve for x

$x = \frac{62^\circ}{2}$

Answer:

$31$