Question

ps bisects ∠qpr. complete the proof that ∠r ≅ ∠q.
statement
1 ps bisects ∠qpr
2 pq ≅ pr
3 ∠qps ≅ ∠rps
4 ps ≅ ps
5 △pqs ≅ △prs
6 ∠r ≅ ∠q
reason
given
reflexive property of congruence
sas
cpctc

Explanation:

Step1: Analyze the given information

We are given that \( \overrightarrow{PS}\) bisects \( \angle QPR\), so \( \angle QPS=\angle RPS\). Also, \( \overline{PQ}\cong\overline{PR}\) (given), and \( \overline{PS}\cong\overline{PS}\) (Reflexive Property of Congruence).

Step2: Use the SAS (Side - Angle - Side) congruence criterion

For two triangles \( \triangle PQS\) and \( \triangle PRS\), we have:

• Side: \( \overline{PQ}\cong\overline{PR}\)
• Angle: \( \angle QPS\cong\angle RPS\) (since \( \overrightarrow{PS}\) bisects \( \angle QPR\))
• Side: \( \overline{PS}\cong\overline{PS}\)

By the SAS (Side - Angle - Side) congruence criterion, \( \triangle PQS\cong\triangle PRS\).

Step3: Use CPCTC (Corresponding Parts of Congruent Triangles are Congruent)

Since \( \triangle PQS\cong\triangle PRS\), then \( \angle R\cong\angle Q\) (by CPCTC).

Answer:

The reason for \( \triangle PQS\cong\triangle PRS\) is SAS (Side - Angle - Side) congruence criterion. And since \( \triangle PQS\cong\triangle PRS\), by CPCTC (Corresponding Parts of Congruent Triangles are Congruent), \( \angle R\cong\angle Q\).