1. **State the problem:** Complete the proof that the lines $\overleftrightarrow{TV}$ and $\overleftrightarrow{QS}$ are parallel. 2. **Given:** $\angle VUW \cong \angle PRQ$ (Given) 3. **Vertical Angle Theorem:** $\angle PRQ \cong \angle SRU$ (Vertical Angle Theorem) 4. **Transitive Property of Congruence:** Since $\angle VUW \cong \angle PRQ$ and $\angle PRQ \cong \angle SRU$, then $\angle VUW \cong \angle SRU$. 5. **Corresponding Angles Postulate:** Angles $\angle VUW$ and $\angle SRU$ are corresponding angles formed by the transversal $\overleftrightarrow{UR}$ intersecting lines $\overleftrightarrow{TV}$ and $\overleftrightarrow{QS}$. 6. **Conclusion:** Since corresponding angles are congruent, by the Corresponding Angles Postulate, the lines $\overleftrightarrow{TV}$ and $\overleftrightarrow{QS}$ are parallel. Thus, $\boxed{\overleftrightarrow{TV} \parallel \overleftrightarrow{QS}}$.