Points A, B, C and D lie on a circle of radius 1. Let x be

Responding to a pm:

A, B, C and D could lie anywhere on the circle. There is nothing given to say that they must lie in that order.
What is the relevance of \(x \lt \pi\) and \(y \lt \pi\)? The total circumference of the circle of radius 1 is \(2\pi\). So this is to tell you that both arcs are less than semi circles so we are talking about the minor arcs of AB and CD.

Question: Is x > y?

1. \(\angle\) ADB is acute

This tells us that D lies on the major arc of AB, not on minor arc i.e. it doesn't lie on the red part which is x. Had D been on x, the angle ADB would have been obtuse.



But y may be smaller than x or greater depending on where C is. Hence this statement alone is not sufficient.

2. \(\angle\) ADB > \(\angle\) CAD
Angles ADB and CAD could be inscribed angles of arcs x and y on their major arcs in which case this implies that x > y. Or angle ADB could be obtuse and arc x could be less than arc y.



Note that in both the cases above, angle ADB > angle CAD. Angle ADB is obtuse since the inscribed angle is in the minor arc. Angle CAD is acute since the angle is in the major arc. But in the first case x > y and in the second case x < y. Hence this statement alone is not sufficient.

Using both statements together, angle ADB is acute and greater than angle CAD so angle CAD is also acute.
This means we are talking about inscribed angles in the major arc in both cases i.e. the first case in the figure given above. Then, if inscribed angle of x is greater than inscribed angle of y, it means central angle of x is greater than central angle of y (Central angle = 2*Inscribed angle in major arc) and hence arc x is greater than arc y.

Answer (C)