Question

angela is constructing the following triangles and draws \\( \overline { vr } \\) parallel to \\( \overline { st } \\).
according to the sas theorem, which of the following needs to be true to show \\( \triangle vrt \cong \triangle str \\)?
\\( \bigcirc \\) a. \\( \overline { vr } \cong \overline { st } \\)
\\( \bigcirc \\) b. \\( \overline { vr } \cong \overline { st } \\) and \\( \angle rvt \cong \angle srt \\)
\\( \bigcirc \\) c. \\( \overline { tv } \cong \overline { rs } \\)
\\( \bigcirc \\) d. \\( \angle rvt \cong \angle tsr \\)

Explanation:

Step1: Recall the SAS (Side - Angle - Side) theorem

The SAS theorem states that if two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, then the two triangles are congruent.

Step2: Analyze the given triangles \(\triangle VRT\) and \(\triangle STR\)

We know that \(RT = RT\) (common side). Since \(VR\parallel ST\), we can use the properties of parallel lines. If \(VR\parallel ST\), then \(\angle VRT=\angle STR\) (alternate interior angles). For the SAS theorem, we need another pair of congruent sides. That pair of sides should be \(VR\) and \(ST\).

Answer:

A. \(\overline{VR}\cong\overline{ST}\)