Question

in the diagram, \\(\frac{sq}{om} = \frac{sr}{on} = 4\\).
to prove that the triangles are similar by the sss similarity theorem, which other sides or angles should be used?
\\(\overline{mn}\\) and \\(\overline{sr}\\)
\\(\overline{mn}\\) and \\(\overline{qr}\\)
\\(\angle s \cong \angle n\\)
\\(\angle s \cong \angle o\\)

Explanation:

Step1: Recall SSS Similarity

SSS similarity needs all three sides in proportion. We know \(\frac{SQ}{OM} = \frac{SR}{ON}=4\). Now check the third pair.

Step2: Calculate Proportions

First, find lengths: \(OM = 15\), \(SQ = 60\); \(ON = 8\), \(SR = 32\); \(MN = 12\), \(QR = 48\). Check \(\frac{QR}{MN}=\frac{48}{12} = 4\). So \(\frac{SQ}{OM}=\frac{SR}{ON}=\frac{QR}{MN}=4\), so sides \(MN\) and \(QR\) are needed.

Answer:

\(\overline{MN}\) and \(\overline{QR}\) (the second option)