The segments MR and NQ meet at point P, where NP = PQ and the angle MNP is

The segments MR and NQ meet at point P, where NP = PQ and the angle MNP is equal to the angle RQP. Prove that MN = RQ.

Let’s complete the drawing. (See the picture on the hosting) Since the angle MNP is equal to the angle PQR, the straight line NQ is a secant with parallel MN and RQ. Then angle NMP = angle PRQ. Since NP = PQ by hypothesis, the triangles MNP and PRQ are equal in three angles and sides, and hence MN = RQ, as required.