Question

given: \\(\overline{sp}\\) bisects \\(\angle rpq\\)
e. \\(\overline{rp} \cong \overline{qp}\\)
prove: \\(\triangle srp \cong \triangle sqp\\)?

Explanation:

Step1: Identify Given Information

We know \( \overline{SP} \) bisects \( \angle RPQ \), so \( \angle RPS \cong \angle QPS \) (Definition of Angle Bisector). Also, \( \overline{RP} \cong \overline{QP} \) (Given), and \( \overline{SP} \cong \overline{SP} \) (Reflexive Property of Congruence).

Step2: Apply SAS Congruence Criterion

In \( \triangle SRP \) and \( \triangle SQP \):

• \( \overline{RP} \cong \overline{QP} \) (Given)
• \( \angle RPS \cong \angle QPS \) (Angle Bisector)
• \( \overline{SP} \cong \overline{SP} \) (Reflexive)

By the Side - Angle - Side (SAS) Congruence Postulate, \( \triangle SRP \cong \triangle SQP \).

Answer:

\( \triangle SRP \cong \triangle SQP \) by SAS Congruence Postulate.