UPSC CDS 2 2022 Math Paper

Concept:A cyclic quadrilateral has opposite angles summing to $180^{\circ}$.In a trapezium, interior angles on the same side of a transversal sum to $180^{\circ}$.Explanation:Since ABCD is cyclic, $angle A + angle C = 180^{\circ}$ (opposite angles).Thus statement 2 is correct.Given $AB \parallel DC$, consider transversal AD.Then $angle A + angle D = 180^{\circ}$ (interior angles on same side).Thus statement 3 is correct.Now, from cyclic property: $angle A + angle C = 180^{\circ}$ and $angle B + angle D = 180^{\circ}$.From parallel property: $angle A + angle D = 180^{\circ}$ and $angle B + angle C = 180^{\circ}$.Comparing gives $angle D = angle C$ and $angle A = angle B$.Hence trapezium ABCD is isosceles, so $AD = BC$.Thus statement 1 is correct.All three statements are true.Answer:D. 1, 2 and 3