Question

plete the proof that \\( \angle r \cong \angle q \\).

statement
1 \\( \overleftrightarrow { p s } \\) bisects \\( \angle q p r \\)
2 \\( \overline { p q } \cong \overline { p r } \\)
3 \\( \angle q p s \cong \angle r p s \\)
4 \\( \overline { p s } \cong \overline { p s } \\)
5 \\( \triangle p q s \cong \triangle p r s \\)
6 \\( \angle r \cong \angle q \\)

reason
given
given
definition of angle bisector
reflexive property of congruence
sas
cpctc

Explanation:

Step1: Use the definition of angle bisector

Since \( \overrightarrow{PS} \) bisects \( \angle QPR \), by the definition of an angle bisector, \( \angle QPS=\angle RPS \).

Step2: Use the reflexive property of congruence

For any segment \( \overline{PS} \), \( \overline{PS}\cong\overline{PS} \) (reflexive property of congruence).

Step3: Apply the SAS (Side - Angle - Side) congruence criterion

We know that \( \overline{PQ}\cong\overline{PR} \) (given), \( \angle QPS\cong\angle RPS \) (from step 1), and \( \overline{PS}\cong\overline{PS} \) (from step 2). So, \( \triangle PQS\cong\triangle PRS \) by the SAS congruence criterion.

Step4: Use CPCTC (Corresponding Parts of Congruent Triangles are Congruent)

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

Answer:

The proof is completed as shown in the steps above, using the angle - bisector definition, reflexive property of congruence, SAS congruence criterion, and CPCTC to show that \( \angle R\cong\angle Q \).