SAT Question 1dc7e423: Lines, Angles, and Triangles answer and explanation with Desmos solution

Choice C is correct. It’s given that the measure of angle $A$ is $52^{\circ }$ and the measure of angle $P$ is $52^{\circ }$. Therefore, angle $A$ is congruent to angle $P$. Choice C states that the measures of angle $B$ and angle $R$ are $32^{\circ }$ and $96^{\circ }$, respectively. The triangle sum theorem states that the measures of the angles in a triangle sum to $180^{\circ }$. In triangle $ABC$, if the measure of angle $B$ is $32^{\circ }$, it follows that the measure of angle $C$ is $(180-(52+32){)}^{\circ }$, or $96^{\circ }$. In triangle $PQR$, if the measure of angle $R$ is $96^{\circ }$, it follows that the measure of angle $Q$ is $(180-(52+96){)}^{\circ }$, or $32^{\circ }$. Therefore, in triangles $ABC$ and $PQR$, angle $B$ is congruent to angle $Q$, and angle $C$ is congruent to angle $R$. If two triangles have three pairs of congruent angles, then the triangles are similar. Therefore, the additional piece of information that is sufficient to prove that triangle $ABC$ is similar to triangle $PQR$ is that the measures of angle $B$ and angle $R$ are $32^{\circ }$ and $96^{\circ }$, respectively.