Question

determining if triangles are congruent
can δtsr and δqrs be proven congruent by sas?
○ yes, because along with the given information on the diagram, $overline{sr} \cong \overline{rs}$ by the reflexive property
○ yes, because a reflection will map δtsr onto δqrs
○ yes, because p appears to be the midpoint of $overline{sq}$ and $overline{tr}$
○ no, because not enough is information given to prove the triangles congruent by sas

Explanation:

To determine if \(\triangle TSR\) and \(\triangle QRS\) are congruent by SAS, we recall 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 triangles are congruent.

Step 1: Identify given congruent sides and angles

• We are given that \(TS = QR=5\) inches (from the diagram, the vertical sides of the two triangles are both 5 inches).
• We are also given that \(\angle T=\angle Q = 66^{\circ}\) (the angles at vertices \(T\) and \(Q\) are both \(66^{\circ}\)).
• The side \(SR\) is common to both \(\triangle TSR\) and \(\triangle QRS\). By the reflexive property of congruence, \(\overline{SR}\cong\overline{RS}\) (a side is congruent to itself).

Step 2: Check the SAS criteria

For \(\triangle TSR\) and \(\triangle QRS\):

• Side \(TS\cong QR\) (given, both 5 inches).
• Angle \(\angle T\cong\angle Q\) (given, both \(66^{\circ}\)).
• Side \(SR\cong RS\) (reflexive property).

So, in \(\triangle TSR\), the sides \(TS\) and \(SR\) with included angle \(\angle T\), and in \(\triangle QRS\), the sides \(QR\) and \(RS\) with included angle \(\angle Q\) satisfy the SAS congruence criterion. The first option correctly states that we can use the given information (the equal sides \(TS = QR\), equal angles \(\angle T=\angle Q\)) and the reflexive property for \(SR\cong RS\) to prove congruence by SAS.

The second option talks about a reflection, but the question is specifically about proving congruence by SAS, not by a reflection (which is a transformation - based congruence, not SAS). The third option assumes \(P\) is a mid - point, but we don't have enough information to confirm that from the given diagram for the purpose of SAS. The fourth option is incorrect because we do have enough information (two sides and the included angle with the help of the reflexive property) to apply SAS.

Answer:

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