SAT Question 7ce2e728: Lines, Angles, and Triangles answer and explanation with Desmos solution

Choice B is correct. It’s given that in triangles $ABC$ and $FGH$, angles $B$ and $G$ each measure $51^{\circ }$, the length of $\overline{AB}$ is $6$, and the length of $\overline{FG}$ is $18$. For statement I, if the measure of angle $A$ is equal to the measure of angle $F$, then triangles $ABC$ and $FGH$ have two pairs of congruent angles, angles $A$ and $F$ and angles $B$ and $G$. When $2$ pairs of angles in $2$ triangles are congruent, the triangles are similar, so statement I is sufficient. For statement II, if the measure of angle $C$ is equal to the measure of angle $H$, then triangles $ABC$ and $FGH$ have $2$ pairs of congruent angles, angles $C$ and $H$ and angles $B$ and $G$. When $2$ pairs of angles in $2$ triangles are congruent, the triangles are similar, so statement II is sufficient. For statement III, if the length of $\overline{FH}$ is $3$ times the length of $\overline{AC}$, then the ratio of the length of $\overline{FH}$ to the length of $\overline{AC}$ is $3$. It’s given that the length of $\overline{FG}$ is $18$ and the length of $\overline{AB}$ is $6$. Therefore, the ratio of the length of $\overline{FG}$ to the length of $\overline{AB}$ is also $3$, so there are two pairs of sides in proportion. However, angle $B$ isn’t between sides $AB$ and $AC$, and angle $G$ isn’t between sides $FG$ and $FH$. Since the given angle measures aren’t between the proportional sides, this information isn't sufficient to prove that the triangles are similar. Therefore, statements I and II are each sufficient, but statement III isn't sufficient.