Question

I need help knowing if Feliz is correct or not it says he’s not convinced that opposite angles of cyclic quadrilateral always add up to 180 degrees. If he moved point G to a different spot on the circle angle FGH would change but angle FEH would still be 66 degrees and these angles would no longer add up to be == 180 degrees . What would be my first step to this review question? And is feliz correct or not?

Answer 1

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:

`\begin{gathered} z=2x,z=2y \\ \therefore \\ x=(z)/(2) \\ y=(z)/(2) \\ Hence \\ x=y \end{gathered}`

This proves that the angle FGH would remain unchanged which makes the opposite angles of the cyclic quadrilateral retain their supplementary property.