Answer
Proof for the problem:
1. $\overline{MN}\parallel\overline{QR}$ (1. Given)
2. $\angle M\cong\angle R$ and $\angle N\cong\angle Q$ (2. The corresponding alternate interior angles for 2 parallel lines must be congruent)
3. $\overline{MN}\cong\overline{QR}$ (3. Given)
4. $\triangle MNP\cong\triangle RQP$ (4. ASA)
Work Step by Step
1) First, it is given that $\overline{MN}\parallel\overline{QR}$
Therefore, the corresponding alternate interior angles for $\overline{MN}$ and $\overline{QR}$ must be congruent.
That means $\angle M\cong\angle R$ and $\angle N\cong\angle Q$
2) It is also given that $\overline{MN}\cong\overline{QR}$
Now we see that 2 angles and the included side of $\triangle MNP$ are congruent with 2 corresponding angles and the included side of $\triangle RQP$.
So we would use ASA to prove triangles congruent.
Now we would construct a proof for the problem:
1. $\overline{MN}\parallel\overline{QR}$ (1. Given)
2. $\angle M\cong\angle R$ and $\angle N\cong\angle Q$ (2. The corresponding alternate interior angles for 2 parallel lines must be congruent)
3. $\overline{MN}\cong\overline{QR}$ (3. Given)
4. $\triangle MNP\cong\triangle RQP$ (4. ASA)
