Question

3.) prove that triangles pqs and rsq are congruent.
given:
\\( \overline { p q } \cong \overline { r s } \\), and \\( \angle p q s \cong \angle r s q \\)

Explanation:

Step1: Identify common side

In triangles \(PQS\) and \(RSQ\), \(\overline{QS}\cong\overline{SQ}\) (common side, reflexive property of congruence).

Step2: Use SAS congruence criterion

We are given \(\overline{PQ}\cong\overline{RS}\) and \(\angle PQS\cong\angle RSQ\).
By the Side - Angle - Side (SAS) congruence criterion (\(PQ = RS\), \(\angle PQS=\angle RSQ\), \(QS = SQ\)), \(\triangle PQS\cong\triangle RSQ\).

Answer:

\(\triangle PQS\cong\triangle RSQ\) by the SAS (Side - Angle - Side) congruence criterion.