Question

δlmn and δpqr are similar triangles. which of the following must be true? ∠n ≅ ∠r ∠m ≅ ∠r ∠n ≅ ∠p ∠m ≅ ∠p

Explanation:

Step1: Find the ratio of corresponding sides

For similar triangles $\triangle LMN$ and $\triangle PQR$, the ratio of sides is $\frac{LM}{PQ}=\frac{12}{8}=\frac{3}{2}$, $\frac{MN}{QR}=\frac{9}{6}=\frac{3}{2}$, $\frac{LN}{PR}=\frac{18}{12}=\frac{3}{2}$. So $\triangle LMN\sim\triangle PQR$ by SSS (Side - Side - Side) similarity criterion.

Step2: Determine corresponding angles

In similar triangles, corresponding angles are congruent. Since $\triangle LMN\sim\triangle PQR$, $\angle M$ corresponds to $\angle Q$, $\angle N$ corresponds to $\angle R$, and $\angle L$ corresponds to $\angle P$.

Answer:

$\angle N\cong\angle R$