Question

can \\( \triangle tsr \\) and \\( \triangle qrs \\) be proven congruent by sas?
× yes, because along with the given information on
the diagram, \\( \overline { s r } \cong r s \\) by the reflexive property
o yes, because a reflection will map \\( \triangle tsr \\) onto
\\( \triangle qrs \\)
o yes, because \\( p \\) appears to be the midpoint of
\\( \overline { s q } \\) and \\( \overline { t r } \\)
o no, because not enough is information given to
prove the triangles congruent by sas

Explanation:

Step1: Recall the SAS (Side - Angle - Side) congruence criterion

For two triangles \(\triangle TSR\) and \(\triangle QRS\), in \(\triangle TSR\) and \(\triangle QRS\), we have \(TS = QR=5\) in (given), \(\angle TSR=\angle QRS = 66^{\circ}\) (given), and \(SR = RS\) (reflexive property of congruence: a segment is congruent to itself).

Step2: Apply the SAS criterion

By the SAS (Side - Angle - Side) congruence criterion, 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. Here, in \(\triangle TSR\) and \(\triangle QRS\), \(TS = QR\), \(\angle TSR=\angle QRS\), and \(SR = RS\). So, \(\triangle TSR\cong\triangle QRS\) by SAS.

Answer:

yes, because along with the given information on the diagram, \(\overline{SR}\cong\overline{RS}\) by the reflexive property