Question

what is ( mangle npq )?
a. 62
b. 116
c. 118

Explanation:

Step1: Identify the parallelogram

First, note that $NP \parallel MQ$ and $MN = PQ$, so $MNPQ$ is a parallelogram. Thus, $MN \parallel PQ$, and $\angle NMP = \angle MPQ = 32^\circ$. Also, $\triangle MNR \cong \triangle PQR$, so $\angle NMR = \angle QPR = 30^\circ$.

Step2: Find $\angle MPQ$

$\angle MPQ = \angle QPR + \angle RPM = 30^\circ + 32^\circ = 62^\circ$

Step3: Calculate $\angle NPQ$

In parallelogram $MNPQ$, $NP \parallel MQ$, so $\angle PNM + \angle NMQ = 180^\circ$. But we use the fact that $\angle NPQ + \angle MPQ = 180^\circ$ (same-side interior angles are supplementary).
$\angle NPQ = 180^\circ - 62^\circ = 118^\circ$

Answer:

C. 118