To find the measure of angle \( m\angle RWU \) given that \( m\angle PVT = 109^\circ \) and that lines \( PQ \) and \( RS \) are parallel, we can use the properties of angles formed by parallel lines cut by a transversal.
- Since \( PQ \parallel RS \) and \( VT \) is the transversal, the angles \( \angle PVT \) and \( \angle RWU \) are corresponding angles.
- Corresponding angles are equal when two parallel lines are cut by a transversal.
Given \( m\angle PVT = 109^\circ \), it follows that \( m\angle RWU = m\angle PVT \).
Thus, we have:
\[ m\angle RWU = 109^\circ \]
The answer is:
109°