Question

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

Explanation:

Identify the proportional sides

Using the SAS Similarity Theorem knowledge point

$$ \frac{LN}{XZ} = \frac{2}{6} = \frac{1}{3} $$

$$ \frac{NM}{ZY} = \frac{3}{9} = \frac{1}{3} $$

Thus, the two pairs of proportional sides are \(LN\) with \(XZ\), and \(NM\) with \(ZY\).

Determine the included angle

Using the Included Angle knowledge point

• For \(\triangle LMN\), the angle included between sides \(LN\) and \(NM\) is \(\angle N\).
• For \(\triangle XYZ\), the angle included between sides \(XZ\) and \(ZY\) is \(\angle Z\).

Apply the similarity requirement

Using the SAS Similarity Theorem knowledge point

$$ \angle N \cong \angle Z $$

Answer:

• \(\angle N \cong \angle Z\) (Correct answer)
• \(\angle N \cong \angle X\)
• \(\angle L \cong \angle Z\)
• \(\angle L \cong \angle Y\)