Question

in the diagram, \\( \frac { s q } { o m } = \frac { s r } { o n } = 4 \\).
to prove that the triangles are similar by the sss similarity theorem, which other sides or angles should be used?
\\( \bigcirc \overline { m n } \\) and \\( \overline { s r } \\)
\\( \bigcirc \overline { m n } \\) and \\( \overline { q r } \\)
\\( \bigcirc \angle s \cong \angle n \\)
\\( \bigcirc \angle s \cong \angle o \\)

Explanation:

Step1: Recall SSS similarity theorem

SSS similarity theorem states that if the ratios of the corresponding sides of two triangles are equal, then the triangles are similar.

Step2: Check the ratios of given sides

We are given \(\frac{SQ}{OM}=4\) (\(SQ = 60\), \(OM=15\), \(\frac{60}{15}=4\)) and \(\frac{SR}{ON}=4\) (\(SR = 32\), \(ON = 8\), \(\frac{32}{8}=4\)). For SSS similarity, we need the ratio of the third pair of corresponding sides.
The third pair of sides should be \(MN\) and \(QR\). \(MN = 12\), \(QR=48\), \(\frac{QR}{MN}=\frac{48}{12}=4\)

Answer:

\(\overline{MN}\) and \(\overline{QR}\)