SAT Question 947a3cde: Lines, Angles, and Triangles answer and explanation with Desmos solution

It is given that the measure of $\angle QPR$ is $60^{\circ }$. Angle $MPR$ and $\angle QPR$ are collinear and therefore are supplementary angles. This means that the sum of the two angle measures is $180^{\circ }$, and so the measure of $\angle MPR$ is $120^{\circ }$. The sum of the angles in a triangle is $180^{\circ }$. Subtracting the measure of $\angle MPR$ from $180^{\circ }$ yields the sum of the other angles in triangle $MPR$. Since $180-120=60$, the sum of the measures of $\angle QMR$ and $\angle NRM$ is $60^{\circ }$. It is given that $MP=PR$, so it follows that triangle $MPR$ is isosceles. Therefore $\angle QMR$ and $\angle NRM$ must be congruent. Since the sum of the measure of these two angles is $60^{\circ }$, it follows that the measure of each angle is $30^{\circ }$. An alternate approach would be to use the exterior angle theorem, noting that the measure of $\angle QPR$ is equal to the sum of the measures of $\angle QMR$ and $\angle NRM$. Since both angles are equal, each of them has a measure of $30^{\circ }$.