Question

which piece of additional information can be used to prove that \\( \triangle rst - \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 \( \triangle RST \) and \( \triangle VUT \), we know \( \angle RTS=\angle VTU \) (vertical angles are congruent).

Step2: Analyze each option

• Option A: \( \overline{RT}=\overline{ST} \) gives information about sides of \( \triangle RST \) only, not related to similarity with \( \triangle VUT \).
• Option B: \( 3ST = UT \) gives a side - length ratio but no information about angles.
• Option C: If \( \angle R\cong\angle V \), then by the AA (Angle - Angle) similarity criterion (\( \angle R\cong\angle V\) and \( \angle RTS=\angle VTU\)), \( \triangle RST\sim\triangle VUT \).
• Option D: \( \angle V\cong\angle U \) gives information about angles of \( \triangle VUT \) only, not related to similarity with \( \triangle RST \).

Answer:

C. \( \angle R\cong\angle V \)