Question

In the figure, quadrilateral W X Y Z is inscribed in circle C. Diagonals segment W Y and segment X Z intersect at the point C, the center of the circle.





Question
Which statements are sufficient to conclude angle X C Y ,begin emphasis,must,end emphasis, be a right angle?



Choose ,begin emphasis,all,end emphasis, sufficient statements.

Answer options with 6 options
A.
the measure of angle X Z Y equals 45°

B.
the measure of angle X Y Z equals 90°

C.
the measure of arc X Y equals 90°

D.
the measure of arc W Z Y equals 180°

E.
the measure of arc W X equals the measure of arc X Y

F.
the measure of arc X Y equals the measure of arc Z W

Answer 1

A, C, E

Brief justifications:
- A: ∠XZY is an inscribed angle intercepting arc XY. If ∠XZY = 45°, then arc XY = 2·45° = 90°, so central ∠XCY = 90°.
- C: The central angle ∠XCY equals the measure of arc XY; if arc XY = 90° then ∠XCY = 90°.
- E: WY is a diameter (diagonal through the center), so arc WX + arc XY = 180°. If arc WX = arc XY then each = 90°, hence ∠XCY = 90°.

(B, D, F are not sufficient — they are true from the given configuration or do not force arc XY to be 90°.)