In the given figure,if $PQ \parallel RS$ and $RS \parallel TU$,and $y: z = 7: 8$,find $x$.

$(96^{\circ})$ Given: $PQ \parallel RS$ and $RS \parallel TU$.
Since lines parallel to the same line are parallel to each other,we have $PQ \parallel TU$.
Therefore,$x = z$ (Alternate interior angles).
Now,for $PQ \parallel RS$,the interior angles on the same side of the transversal are supplementary.
Therefore,$x + y = 180^{\circ}$.
Substituting $x = z$ in the equation,we get $z + y = 180^{\circ}$.
Given the ratio $y: z = 7: 8$,let $y = 7k$ and $z = 8k$.
Substituting these into the equation: $7k + 8k = 180^{\circ} \implies 15k = 180^{\circ} \implies k = 12^{\circ}$.
Thus,$y = 7 \times 12^{\circ} = 84^{\circ}$ and $z = 8 \times 12^{\circ} = 96^{\circ}$.
Since $x = z$,we have $x = 96^{\circ}$.