We are given that ray $VX$ bisects angle $WVY$ and ray $VY$ bisects angle $XVZ$. We need to prove that angle $WVX$ is congruent to angle $YVZ$, which means $\angle WVX \cong \angle YVZ$.

1. Problem Description

We are given that ray $VX$ bisects angle $WVY$ and ray $VY$ bisects angle $XVZ$. We need to prove that angle $WVX$ is congruent to angle $YVZ$, which means $\angle WVX \cong \angle YVZ$.

2. Solution Steps

Here is the two-column proof:

| Statements | Reasons |

|---|---|

|

1. $\overrightarrow{VX}$ bisects $\angle WVY$ |

1. Given |

|

2. $\angle WVX \cong \angle XVY$ |

2. Definition of angle bisector |

|

3. $\overrightarrow{VY}$ bisects $\angle XVZ$ |

3. Given |

|

4. $\angle XVY \cong \angle YVZ$ |

4. Definition of angle bisector |

|

5. $\angle WVX \cong \angle YVZ$ |

5. Transitive Property of Congruence |

The transitive property of congruence states that if $a \cong b$ and $b \cong c$, then $a \cong c$. In this case, $\angle WVX \cong \angle XVY$ and $\angle XVY \cong \angle YVZ$, so $\angle WVX \cong \angle YVZ$.

3. Final Answer

$\angle WVX \cong \angle YVZ$