Question

in the diagram, the ratios of two pairs of corresponding sides are equal to prove that \\( \triangle lmn - \triangle xyz \\) by the sas similarity theorem, it also needs to be shown that \\( \angle n \cong \angle z \\) \\( \angle n \cong \angle x \\) \\( \angle l \cong \angle z \\) \\( \angle l \cong \angle y \\)

Explanation:

Step1: Recall SAS similarity theorem

Two triangles are similar by SAS (Side - Angle - Side) similarity theorem if the ratios of two pairs of corresponding sides are equal and the included angles are congruent.

Step2: Identify corresponding sides and their ratios

For \( \triangle LMN \) and \( \triangle XYZ \), we have \( \frac{LN}{XZ}=\frac{2}{6}=\frac{1}{3} \) and \( \frac{MN}{YZ}=\frac{3}{9}=\frac{1}{3} \).

Step3: Determine the included angles

The included angle for sides \( LN \) and \( MN \) in \( \triangle LMN \) is \( \angle N \), and the included angle for sides \( XZ \) and \( YZ \) in \( \triangle XYZ \) is \( \angle Z \). So, to apply SAS similarity theorem, we need \( \angle N\cong\angle Z \).

Answer:

A. \( \angle N\cong\angle Z \)