Question

which of the following is a good first step to prove that $\triangle pqr \sim \triangle abc$?
apply a translation to $\triangle pqr$ that takes $p$ to $b$.
apply a translation and rotation to $\triangle pqr$ that maps $\angle q$ onto $\angle a$.
apply a dilation centered at $r$ with scale factor $\frac{ac}{qr}$ to $\triangle pqr$.
apply a reflection across $\overline{pr}$ to $\triangle pqr$.

Explanation:

To prove similarity, we first need to align corresponding congruent angles. We know $\angle Q \cong \angle B$, $\angle P \cong \angle A$, so mapping $\angle Q$ onto $\angle A$ is incorrect. The correct first step is to map the matching congruent angles by translating and rotating $\triangle PQR$ to overlay $\angle Q$ onto $\angle B$. The only valid option that starts aligning corresponding angles properly is the second choice, as it begins to map the congruent marked angles to set up the similarity proof.

Answer:

B. Apply a translation and rotation to $\triangle PQR$ that maps $\angle Q$ onto $\angle A$.