Question

11/14 - triangle congruence schoology assignment
is △pqr ≅ △psr? explain
choose the correct answer below
○ 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.
○ a.
○ b.
○ c.
○ d.

Explanation:

Step1: Check congruent sides and angles

We know that \( \overline{PQ}\cong\overline{PS}\) (given by the marks on the sides), \( \overline{PR}\cong\overline{PR}\) (common side). Also, \( \angle QPR\cong\angle SPR\) (given by the angle - marking).

Step2: Apply congruence criteria

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.
For \(\triangle PQR\) and \(\triangle PSR\), we have two sides (\(\overline{PQ}\cong\overline{PS}\), \(\overline{PR}\cong\overline{PR}\)) and the included angle (\(\angle QPR\cong\angle SPR\)) 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.