Let A and B be two points on a circle with centre O.

(A) Are there points X, Y on the circle, on the same side of AB, such that ∠AXB is different from ∠AYB?

(B) Is it true that if ∠AXB = ∠AYB, then X and Y lie on the same side of the circle?

(C) If ∠AXB = ∠AYB, and X and Y do not lie on the circle, does the circle through A, B and X also pass through Y?                                                                                                                                                                                                               [Page No. 111]

(A) No

Reason:

If X and Y lie on the same side of chord AB, then they lie on the same arc AB.

Angles subtended by the same chord in the same segment of a circle are equal.

Therefore, $\angle AXB = \angle AYB$

So, there are no such point X and Y on the same side of AB for which the angles are different.

(B) Not necessarily.
X and Y may lie on opposite sides of AB with ∠AXB = ∠AYB if each angle equals 90° (if AB is a diameter) or more generally, on opposite arcs the subtended angles are supplementary (they add up to 180°).
So, it is possible that ∠AXB = ∠AYB when each equal 90°, even if they lie on opposite sides.

However, whenever they are on the same side, the angles are equal.

So, the statement is false in general.

(C) Yes. If X and Y lie on the same side of AB and $\angle AXB = \angle AYB$, then by Theorem (the converse of the angles-in-same-segment result), A, B, X, Y are concyclic. So, the circle through A, B, X also passes through Y.