10.4. In pentagon $M N P Q S: \quad M N=N P=P Q=Q S=S M$ and $\angle M N P=2 \angle Q N S$. Find the measure of angle $M N P$.
Answer: $60^{\circ}$

Solution. Since $\angle S N Q=\angle M N S+\angle P N Q$, we can take a point $T$ on the side $S Q$ such that $\angle S N T=\angle M N S=\angle M S N$, i.e., $N T \| M S$.

Then $\angle T N Q=\angle S N Q-\angle S N T=\angle P N Q=\angle N Q P$, i.e., $N T \| P Q$. Therefore, $M S \| P Q$, and since $M S=P Q$, $P Q S M$ is a parallelogram. Thus, $M P=S Q$, which means triangle $M N P$ is equilateral and $\angle M N P=60^{\circ}$.