In the figure above, and intersect at point P, NP = QP, and MP = PR. What is the measure, in degrees, of ∠QMR ? (Disregard the degree symbol when gridding your answer.)
It is given that the measure of ∠QPR is 60°. Angle MPR and ∠QPR are collinear and therefore are supplementaryangles. This means that the sum of the two angle measures is 180°, and so the measure of ∠MPR is 120°. The sum of t he angles in a triangle is 180°. Subtracting the measure of ∠MPR from 180° yields the sum of the other angles in the triangle MPR. Since 180 − 120 = 60, the sum of the measures of ∠QMR and ∠NRM is 60°. It is given thatMP = PR, so it follows that triangle MPR is isosceles. Therefore ∠QMR and ∠NRM must be congruent. Since t he sum of the measure of these two angles is 60°, it follows that the measure of each angle is 30°. An alternate approach would be to use the exterior angle theorem, noting that the measure of ∠QPR is equal to the sum of the measures of ∠QMR and ∠NRM. Since both angles are equal, each of them has a measure of 30°.