1. **State the problem:** We are given two parallel lines \(\overleftrightarrow{PQ} \parallel \overleftrightarrow{RS}\) and a transversal \(\overleftrightarrow{TU}\) intersecting them at points \(V\) and \(W\). We want to prove that the corresponding angles \(\angle RWV\) and \(\angle PVT\) are congruent. 2. **Given:** \(\overleftrightarrow{PQ} \parallel \overleftrightarrow{RS}\). 3. **Step 2 in proof:** \(\angle RWV \cong \angle QVW\) because they are vertical angles formed by the intersection of the transversal \(\overleftrightarrow{TU}\) with the line \(\overleftrightarrow{RS}\). 4. **Step 3 in proof:** \(\angle QVW \cong \angle PVT\) because they are corresponding angles formed by the transversal \(\overleftrightarrow{TU}\) intersecting the parallel lines \(\overleftrightarrow{PQ}\) and \(\overleftrightarrow{RS}\). 5. **Step 4 in proof:** By the Transitive Property of Congruence, since \(\angle RWV \cong \angle QVW\) and \(\angle QVW \cong \angle PVT\), it follows that \(\angle RWV \cong \angle PVT\). **Summary:** - Vertical angles are congruent. - Corresponding angles formed by a transversal crossing parallel lines are congruent. - Using these facts and transitivity, the desired angle congruence is proven. \[\boxed{\angle RWV \cong \angle PVT}\]