The quadrilateral ABCD is cyclic. The opposite angles of a cyclic quadrilateral are supplementary. So, x + 100° = 180° Therefore, x = 80°.
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Yes. If ∠AXB = ∠AYB, then A, B, X and Y are concyclic, provided X and Y are on the same side of AB. This follows from Theorem 10.
Yes, provided X and Y lie on the same side of AB. Angles subtended by the same chord at points on the same segment of a circle are equal.
Yes. If X and Y lie on the same side of AB on the circle, then ∠AXB and ∠AYB subtend the same chord AB. Hence, they are equal, so they cannot be different.
Since OA and OB are radii, triangle AOB is isosceles. Given ∠AOB = 60°, the other two angles are also 60°. Thus, triangle AOB is equilateral, so AB = OA = 12 cm.