Let $t$ be TNYWR.

In the diagram, line segments $A B$ and $C D$ are parallel. $A B$ intersects $E F$ at $V$ and $G F$ at $W$. $C D$ intersects $E F$ at $Y$ and $G F$ at $Z$. If $\angle A V E=72^{\circ}, \angle E F G=t^{\circ}$, and $\angle G Z D=x^{\circ}$, what is the value of $x$ ?

![](https://cdn.mathpix.com/cropped/2024_04_30_4efeb798298dfec26b46g-4.jpg?height=505&width=721&top_left_y=951&top_left_x=732)
Since $A B$ and $C D$ are parallel, then $\angle F Y Z=\angle A V Y=72^{\circ}$ (alternating angles).

Also, $\angle F Z Y=\angle D Z G=x^{\circ}$ (opposite angles).

In $\triangle F Y Z$, the sum of the angles is $180^{\circ}$, and so $\angle Y F Z+\angle F Y Z+\angle F Z Y=180^{\circ}$.

Thus, $t^{\circ}+72^{\circ}+x^{\circ}=180^{\circ}$ or $x=108-t$.

Since the answer to (a) is 52 , then $t=52$ and so $x=108-t=56$.