Given that lines TV and WY are parallel, we can use properties of parallel lines and transversal angles to find \( m \angle TUS \).
From your description, we can assume the angles are configured such that:
- \( \angle WXZ = 129^\circ \) is an exterior angle, which means it equals the sum of the two opposite interior angles because of the transversal created by line WY.
The angles formed by the transversal line (line WX for example) dividing the parallel lines (TV and WY) can be labeled as:
- \( \angle WXZ \) and \( \angle TUS \)
Since \( TV \) is parallel to \( WY \) and \( WX \) acts as a transversal, the corresponding angles are equal, so:
\[ m \angle TUS = m \angle WXZ \]
Since we know \( m \angle WXZ = 129^\circ \), we find:
\[ m \angle TUS = 129^\circ \]
Thus, the measure of angle \( TUS \) is \( \boxed{129^\circ} \).