Question

lmn and \\( \triangle pqr \\) are similar triangles. which of the following must be true? \\( \angle n \cong \angle r \\) \\( \angle m \cong \angle r \\) \\( \angle n \cong \angle p \\) \\( \angle m \cong \angle p \\)

Explanation:

Step1: Find the ratio of corresponding sides

For \(\triangle LMN\) and \(\triangle PQR\), \(\frac{LM}{PQ}=\frac{12}{8}=\frac{3}{2}\), \(\frac{MN}{QR}=\frac{9}{6}=\frac{3}{2}\), \(\frac{LN}{PR}=\frac{18}{12}=\frac{3}{2}\). So \(\triangle LMN\sim\triangle PQR\) (by SSS similarity criterion).

Step2: Determine corresponding angles

Since \(\triangle LMN\sim\triangle PQR\), corresponding angles are equal. \(LM\) corresponds to \(PQ\), \(MN\) corresponds to \(QR\), \(LN\) corresponds to \(PR\). So \(\angle M\) corresponds to \(\angle Q\), \(\angle N\) corresponds to \(\angle R\), \(\angle L\) corresponds to \(\angle P\).

Answer:

\(\angle N\cong\angle R\)