Question

in the figure above, (overline{mq}) and (overline{nr}) intersect at point (p), (mp = qp), and (np = rp). what is the measure, in degrees, of (angle qmr)? (disregard the degree symbol when gridding your answer)

Explanation:

Step1: Prove triangle congruence

In $\triangle MAP$ and $\triangle QAP$, we have $MP = QP$, $AP=AP$ (common - side), and $AM = AQ$. By the Side - Side - Side (SSS) congruence criterion, $\triangle MAP\cong\triangle QAP$.

Step2: Find angle measure

Since $\triangle MAP\cong\triangle QAP$, $\angle MAP=\angle QAP$. And $\angle MAQ = 40^{\circ}$, so $\angle MAP=\angle QAP=\frac{1}{2}\angle MAQ$. Then $\angle MAP = 20^{\circ}$.

Answer:

20