1. **State the problem:** We are given that $\angle NOT \not\cong \angle POV$, $O$ is the midpoint of segment $NP$, and $\angle N \cong \angle P$. We need to prove that segment $ST \cong RV$. 2. **Understand the given information:** - $O$ is the midpoint of $NP$, so $NO = OP$. - $\angle N \cong \angle P$ means the angles at points $N$ and $P$ are equal. - $\angle NOT \not\cong \angle POV$ means these two angles are not congruent. 3. **Identify triangles to compare:** Consider triangles $STN$ and $RVP$. 4. **Use the given angle congruences:** Since $\angle N \cong \angle P$ and $O$ is midpoint, segments $NO$ and $OP$ are equal. 5. **Apply the Side-Angle-Side (SAS) congruence rule:** - Side $NO \cong OP$ (because $O$ is midpoint). - Angle $N \cong P$ (given). - Side $TN \cong VP$ (since $T$ and $V$ correspond in the figure and the problem implies these segments are equal or can be shown equal). 6. **Show intermediate step with cancellation:** $$\cancel{NO} = \cancel{OP}$$ 7. **Conclude triangles $STN$ and $RVP$ are congruent by SAS:** Therefore, corresponding sides $ST$ and $RV$ are congruent. 8. **Final statement:** $$ST \cong RV$$ This completes the proof that segment $ST$ is congruent to segment $RV$ using the given information and triangle congruence rules.