Question

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is △pqr≅△psr? explain.
a. yes. pq≅ps, pr≅pr, and ∠qpr≅∠spr. thus, △pqr≅△psr by sas.
b. yes. pq≅ps, pr≅pr, and sr≅qr. thus, △pqr≅△psr by sss.
c. no. although pq≅ps and ∠qpr≅∠spr, sr≇qr. thus, △pqr≇△psr.
d. it cannot be determined. although pq≅ps and ∠qpr≅∠spr, at least one more pair of sides needs to be congruent for the triangles to be congruent.

Explanation:

Step1: Identify congruent parts

From the figure, we can see that $\overline{PQ}\cong\overline{PS}$ (marked with one - dash), $\overline{PR}=\overline{PR}$ (common side), and $\angle QPR\cong\angle SPR$ (marked with one - arc).

Step2: Apply congruence criterion

The Side - Angle - Side (SAS) congruence criterion 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. Here, we have two sides ($\overline{PQ}$ and $\overline{PR}$ in $\triangle PQR$; $\overline{PS}$ and $\overline{PR}$ in $\triangle PSR$) and the included angle ($\angle QPR$ and $\angle SPR$) congruent.

Answer:

A. Yes. $\overline{PQ}\cong\overline{PS}$, $\overline{PR}\cong\overline{PR}$, and $\angle QPR\cong\angle SPR$. Thus, $\triangle PQR\cong\triangle PSR$ by SAS.