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 \overline{MQ} can be completed (the measure of angle MPR must be 120°).

From there, you’re almost home. The question tells you that MPPR, 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°.