1. **State the problem:** Given that lines QS and TV are parallel, prove that $m\angle VUW + m\angle PRS = 180^\circ$. 2. **Recall the given information and theorems:** - $QS \parallel TV$ (Given) - $\angle VUW \cong \angle SRU$ (Corresponding Angles Theorem) - $m\angle PRS + m\angle SRU = 180^\circ$ (Linear pair sum to 180°) 3. **Use the Corresponding Angles Theorem:** Since $\angle VUW \cong \angle SRU$, their measures are equal: $$m\angle VUW = m\angle SRU$$ 4. **Substitute into the linear pair equation:** $$m\angle PRS + m\angle SRU = 180^\circ$$ Replace $m\angle SRU$ with $m\angle VUW$: $$m\angle PRS + m\angle VUW = 180^\circ$$ 5. **Conclusion:** Therefore, the sum of the measures of angles $VUW$ and $PRS$ is $180^\circ$: $$m\angle VUW + m\angle PRS = 180^\circ$$ This completes the proof.