Feliz is correct when he said that opposite angles of a cyclic quadrilateral always add up to 180 degrees.
In the question, if point G is moved, the angle FGH will remain unchanged regardless. Consider the image below:
Recall the rule: "The angle subtended by an arc at the center is double the angle on the circle."
This means that:
[tex]\begin{gathered} z=2x,z=2y \\ \therefore \\ x=\frac{z}{2} \\ y=\frac{z}{2} \\ Hence \\ x=y \end{gathered}[/tex]
This proves that the angle FGH would remain unchanged which makes the opposite angles of the cyclic quadrilateral retain their supplementary property.
