Question

kelly is constructing the following triangles. according to the sss theorem, which of the following needs to be true to show \\( \triangle vrt \cong \triangle str \\)?
○a. \\( \overline { rs } \cong \overline { tv } \\) and \\( \angle rts \cong \angle rvt \\)
○b. \\( \angle srt \cong \angle vtr \\) and \\( \overline { st } \cong \overline { rv } \\)
○c. \\( \overline { vr } \cong \overline { rs } \\) and \\( \overline { tv } \cong \overline { st } \\)
○d. \\( \overline { tv } \cong \overline { rs } \\) and \\( \overline { vr } \cong \overline { st } \\)

Explanation:

Step1: Recall SSS theorem

SSS (Side - Side - Side) theorem states that if three sides of one triangle are congruent to three sides of another triangle, then the two triangles are congruent. For \(\triangle VRT\) and \(\triangle STR\), we already have \(RT = TR\) (common side).

Step2: Analyze each option

• Option A: \(\angle RTS\cong\angle RVT\) is an angle - congruence, not relevant for SSS.
• Option B: \(\angle SRT\cong\angle VTR\) is an angle - congruence, not relevant for SSS.
• Option C: \(VR\cong RS\) and \(TV\cong ST\) does not give the correct side - congruences for \(\triangle VRT\) and \(\triangle STR\) to use SSS.
• Option D: If \(TV\cong RS\) and \(VR\cong ST\), along with \(RT = TR\), then by SSS (\(VR\cong ST\), \(TV\cong RS\), \(RT\cong TR\)), \(\triangle VRT\cong\triangle STR\).

Answer:

D. \(\overline{TV}\cong\overline{RS}\) and \(\overline{VR}\cong\overline{ST}\)