To solve for \( x \) in the problem where \( PQ \parallel RS \), we can use the properties of parallel lines and the angles formed by a transversal.
Given the angles:
- \( \angle UWS = 6x - 18 \)
- \( \angle QVU = 102^\circ \)
Since \( PQ \parallel RS \), and \( UWS \) and \( QVU \) are corresponding angles, we can set them equal to each other:
\[ 6x - 18 = 102 \]
Now, we will solve for \( x \).
- Add 18 to both sides:
\[ 6x = 102 + 18 \] \[ 6x = 120 \]
- Divide both sides by 6:
\[ x = \frac{120}{6} \] \[ x = 20 \]
Thus, the value of \( x \) is \( \boxed{20} \).