Test 7 Section 3 #17
The first thing to do on any find-the-measure-of-a-certain-angle problem is complete any 180°s you can. In this case, triangle PQR can be completed (the measure of angle PRQ must be 50°) and can be completed (the measure of angle MPR must be 120°).
From there, you’re almost home. The question tells you that MP = PR, so you know that triangle MPR is isosceles. That means that angles PMR and PRM must be congruent! Because that triangle already has a 120° angle in it, the two unknowns must add up to 60°. Because they must be congruent, they must each be 30°.
Therefore, the measure of angle QMR is 30°.