#### Answer

- From the given information, we can deduce $\angle 1\cong\angle 2$ and $\angle 3\cong\angle 4$
- Then, prove that $\triangle MQP\cong\triangle PNM$ by method ASA
- Then, by CPCTC, $\overline{MQ}\cong\overline{PN}$

#### Work Step by Step

*PLANNING:
- From the given information, we can deduce $\angle 1\cong\angle 2$ and $\angle 3\cong\angle 4$
- Then, prove that $\triangle MQP\cong\triangle PNM$
- Then, by CPCTC, $\overline{MQ}\cong\overline{PN}$
1) $\overline{MN}\parallel\overline{QP}$ and $\overline{MQ}\parallel\overline{NP}$ (Given)
2) $\angle 1\cong\angle 2$ and $\angle 3\cong\angle 4$ (if 2 lines are parallel, then the alternate interior angles for these 2 lines are congruent)
3) $\overline{MP}\cong\overline{PM}$ (Identity)
So now we have 2 angles and the included side of $\triangle MQP$ are congruent with 2 corresponding angles and the included side of $\triangle PNM$
4) $\triangle MQP\cong\triangle PNM$ (ASA)
5) $\overline{MQ}\cong\overline{PN}$ (CPCTC)