We are given triangle $PQR$ where side $PQ$ is congruent to side $RQ$, i.e., $PQ = RQ$. We need to prove that angle $P$ is congruent to angle $R$, i.e., $\angle P = \angle R$.

1. Problem Description

We are given triangle $PQR$ where side $PQ$ is congruent to side $RQ$, i.e., $PQ = RQ$. We need to prove that angle $P$ is congruent to angle $R$, i.e., $\angle P = \angle R$.

2. Solution Steps

Statements | Reasons

------- | --------

1. $\triangle PQR$ |

1. Given

2. $PQ = RQ$ |

2. Given

3. Draw $QS$ such that $QS$ bisects $\angle PQR$ |

3. Every angle has exactly one bisector.

4. $\angle PQS = \angle RQS$ |

4. Definition of angle bisector

5. $QS = QS$ |

5. Reflexive Property

6. $\triangle PQS \cong \triangle RQS$ |

6. Side Angle Side (SAS) Congruence Theorem ($PQ = RQ$, $\angle PQS = \angle RQS$, $QS = QS$)

7. $\angle P = \angle R$ |

7. Corresponding Parts of Congruent Triangles are Congruent (CPCTC)

3. Final Answer

$\angle P = \angle R$