In the given problem, you have two parallel lines \( TV \) and \( WY \) with the transversal line \( SU \).
You are given that \( \angle TUS = 117^\circ \). We need to find \( m \angle VUX \).
Since \( TV \) and \( WY \) are parallel and \( SU \) is a transversal line, corresponding angles are equal.
Looking at the angles formed by the transversal with the parallel lines:
- \( \angle TUS \) is an exterior angle at line \( TV \).
- \( \angle VUX \) is the corresponding angle at line \( WY \).
Thus, since these angles are corresponding angles, we have: \[ m \angle VUX = m \angle TUS = 117^\circ. \]
Therefore, \[ m \angle VUX = 117^\circ. \]