Triangle Congruence 67549D

1. **State the problem:** We are given that $SQ$ bisects $\angle PSR$ and that $\angle SPQ \cong \angle SRQ$. We need to prove that $PS \cong SR$. 2. **Understand the given information:** - $SQ$ bisects $\angle PSR$ means $\angle PSQ = \angle QSR$. - $\angle SPQ \cong \angle SRQ$ means these two angles are equal. 3. **Identify triangles to compare:** Consider triangles $PSQ$ and $RSQ$. 4. **List known equal parts:** - $SQ$ is common side to both triangles. - $\angle PSQ = \angle QSR$ (since $SQ$ bisects $\angle PSR$). - $\angle SPQ = \angle SRQ$ (given). 5. **Apply Angle-Angle-Side (AAS) congruence criterion:** - Two angles and the non-included side are equal in both triangles. 6. **Conclude triangles are congruent:** $$\triangle PSQ \cong \triangle RSQ$$ 7. **Therefore, corresponding sides are equal:** $$PS \cong SR$$ **Final answer:** $PS \cong SR$ as required.