1. As shown in Figure $3 . \triangle P A B 、 \triangle P C D$ are both equilateral triangles. $M 、 N$ are the midpoints of $P B 、 C D$ respectively, and $P Q \perp A C$ at $Q$. Then $\angle M Q N=$
$=.1 .30^{\circ}$.
Connect $A M, P N$. It is easy to prove that $A, P, M, Q$ are concyclic. Therefore, $\angle P Q M=\angle P A M=30^{\circ}$. Also, $P, N, C, Q$ are concyclic, so we get $\angle C Q N=\angle C P N=30^{\circ}$.