To determine the measure of \( m \angle QRU \), we can use the fact that lines \( QS \) and \( TV \) are parallel and that \( VUW \) and \( QRU \) are corresponding angles.
Given that the measure of angle \( m VUW = 137^\circ \), and since corresponding angles are equal when lines are parallel, we have:
\[ m \angle QRU = m \angle VUW = 137^\circ \]
Thus, the measure of \( m \angle QRU \) is \( 137^\circ \).