In the situation you've described, where \( PQ \parallel RS \) and a transversal \( TV \) intersects these lines, the angles formed can be analyzed using the properties of parallel lines.
Given that \( m\angle PVT = 109^\circ \), we first note that angle \( PVT \) and angle \( RWT \) (which is formed on line \( RS \) by transversal \( TV \)) are corresponding angles.
Since \( PQ \parallel RS \) and are cut by the transversal \( TV \), corresponding angles are equal. Therefore, we can derive that:
\[ m\angle RWT = m\angle PVT = 109^\circ \]
Next, angles \( RWU \) and \( RWT \) are supplementary because they form a linear pair (they are on a straight line). Therefore, we can find \( m\angle RWU \) as follows:
\[ m\angle RWU + m\angle RWT = 180^\circ \] \[ m\angle RWU + 109^\circ = 180^\circ \] \[ m\angle RWU = 180^\circ - 109^\circ = 71^\circ \]
Thus, the measure of angle \( RWU \) is \( 71^\circ \).
The correct answer is: 71°