Question: Complete the proof that $m\angle PRS + m\angle VUW = 180^\circ$.
Statement Reason
1 $XZ \parallel QS$ Given
2 $XZ \parallel TV$ Given
3 $\angle VUW \cong \angle WYZ$
4 $m\angle WYZ + m\angle PRS = 180^\circ$
5 $m\angle PRS + m\angle VUW = 180^\circ$ Substitution
1. **State the problem:**
We need to prove that $$m\angle PRS + m\angle VUW = 180^\circ$$ given the parallel lines and angle relationships.
2. **Given information:**
- $XZ \parallel QS$
- $XZ \parallel TV$
- $\angle VUW \cong \angle WYZ$
- $m\angle WYZ + m\angle PRS = 180^\circ$
3. **Recall the substitution property:**
If two angles are congruent, their measures are equal. So,
$$m\angle VUW = m\angle WYZ$$
4. **Use substitution in the equation:**
Given:
$$m\angle WYZ + m\angle PRS = 180^\circ$$
Substitute $m\angle VUW$ for $m\angle WYZ$:
$$m\angle PRS + m\angle VUW = 180^\circ$$
5. **Conclusion:**
We have shown that the sum of the measures of angles $PRS$ and $VUW$ is $180^\circ$ by substitution, completing the proof.