Question

which piece of additional information can be used to prove that \\( \triangle rst \sim \triangle vut \\)?
\\( \overline { rt } = \overline { st } \\)
\\( 3 st = ut \\)
\\( \angle r \cong \angle v \\)
\\( \angle v \cong \angle u \\)

Explanation:

Step1: Recall similarity criteria

For two triangles to be similar, we can use AA (Angle - Angle) similarity criterion. In \(\triangle RST\) and \(\triangle VUT\), \(\angle RTS=\angle VTU\) (vertically opposite angles). If we have another pair of equal angles, we can prove the similarity.

Step2: Analyze each option

• Option \(\overline{RT}=\overline{ST}\): Only gives information about sides of \(\triangle RST\), not related to similarity of \(\triangle RST\) and \(\triangle VUT\).
• Option \(3ST = UT\): Gives a ratio of sides, but without knowing the ratio of other sides or angles, not sufficient for similarity.
• Option \(\angle R\cong\angle V\): Since \(\angle RTS=\angle VTU\) (vertically opposite angles) and if \(\angle R\cong\angle V\), then by AA (Angle - Angle) similarity criterion \(\triangle RST\sim\triangle VUT\).
• Option \(\angle V\cong\angle U\): Gives information about angles of \(\triangle VUT\), not related to \(\triangle RST\) for similarity.

Answer:

\(\angle R\cong\angle V\)