To find the value of \( x \), we can use the property that when two parallel lines are crossed by a transversal, corresponding angles and alternate interior angles are equal.
In the given scenario, we know that:
- \( PQ \parallel RS \)
- \( \angle UWS = 4x - 18^\circ \)
- \( \angle QVU = 102^\circ \)
Since the lines \( PQ \) and \( RS \) are parallel, and \( UWS \) and \( QVU \) are angles formed by a transversal that crosses them, we can set up the equation:
\[ \angle UWS = \angle QVU \]
Substituting the angles in gives us:
\[ 4x - 18 = 102 \]
Now, solve for \( x \):
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Add 18 to both sides: \[ 4x = 102 + 18 \] \[ 4x = 120 \]
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Divide both sides by 4: \[ x = \frac{120}{4} \] \[ x = 30 \]
Thus, the value of \( x \) is \( \boxed{30} \).