To find the measure of angle \( mQRU \), we will use the properties of parallel lines and the angles formed by a transversal.
Given:
- Lines \( QS \) and \( TV \) are parallel.
- Angle \( mQRP = 131^\circ \).
Since \( QS \) and \( TV \) are parallel, and \( QR \) serves as a transversal, we can apply the concept of alternate interior angles.
Angle \( mQRU \) and angle \( mQRP \) are corresponding angles because they are on the same side of the transversal \( QR \).
Since corresponding angles are equal when two lines are parallel, we have:
\[ mQRU = mQRP \]
Thus:
\[ mQRU = 131^\circ \]
So, the measure of \( mQRU \) is \( 131^\circ \).