1. In the cyclic quadrilateral ABCD, if AB = AD, \(\angle\)DAC = 70° and \(\angle\)BDC = 50°, then what is the measure of \(\angle\)ACD?

(a) 30\(^o\) (b) 40\(^o\) (c) 50\(^o\) (d) 70\(^o\)

2. ABCD is a cyclic quadrilateral, and CD is extended up to point E. If \(\angle\)ADE = 92°, then what is the measure of \(\angle\)ABC?

(a) 88\(^o\) (b) 29\(^o\) (c) 92\(^o\) (d) 60\(^o\)

3. If the ratio of three consecutive angles of a cyclic quadrilateral is 1 : 2 : 3, what are the measures of the first and third angles?

4. AB is extended to point X in the cyclic quadrilateral ABCD. If \(\angle\)XBC = 98° and \(\angle\)ADB = 45°, then what is the measure of \(\angle\)BAC?

5. If the ratio of three consecutive angles of a cyclic quadrilateral is 1:2:3, then what are the measures of the first and third angles?

6. In the cyclic quadrilateral ABCD, AB is the diameter and \(\angle\)ACD = 50°, then what is the measure of \(\angle\)BAD?

(a) 30° (b) 40° (c) 50° (d) 60°

7. In the cyclic quadrilateral ABCD, side AB is extended up to point P. If ∠CAB = 35° and ∠CBP = 82°, then what is the measure of ∠ADB?

(a) 47° (b) 55° (c) 35° (d) 60°

8. ABCD is a cyclic quadrilateral and O is the center of the circle. If \(\angle\)COD = 120° and \(\angle\)BAC = 30°, then what is the measure of \(\angle\)BOC?

9. In the cyclic quadrilateral ABCD, side AB is extended up to point X. If \(\angle\)XBC = 82° and \(\angle\)ADB = 47°, then what is the measure of \(\angle\)BAC?

(a) 45° (b) 45° (c) 35° (d) 60°

10. If ABCD is a cyclic quadrilateral and ∠A = 100°, then the measure of ∠C is—?

(a) 50° (b) 200° (c) 80° (d) 180°

11. The center of a circle is O, and AB is its diameter. ABCD is a cyclic quadrilateral. If \(\angle\)ABC = 65° and \(\angle\)DAC = 40°, then the measure of \(\angle\)BCD is—?

(a) 75° (b) 105° (c) 115° (d) 80°

12. In the cyclic quadrilateral PQRS, the side PS is a diameter of the circle. If \(\angle\)PQR = 120°, then what is the measure of \(\angle\)SPR?

(a) 90° (b) 30° (c) 60° (d) 120°

13. ABCD is a cyclic quadrilateral. If ∠ ABD = 48°, what is the measure of ∠ ACD?

(a) 42° (b) 138° (c) 48° (d) 12°

14. In the cyclic quadrilateral PORS, the side PS is a diameter of the circle. If \(\angle\)PQR = 128°, then what is the measure of \(\angle\)SPR?

(a) 30° (b) 38° (c) 60° (d) None of the above

15. ABCD is a cyclic quadrilateral. If BC is the diameter and ∠ADC = 130°, then what is the value of ∠ACB?

16. If one angle of a cyclic quadrilateral is 75°, what is the measure of its opposite angle?

17. ABCD is a cyclic trapezium in which sides AD and BC are parallel to each other. If \(\angle\)ABC = 75°, then what is the measure of \(\angle\)BCD?

(a) 105° (b) 90° (c) 150° (d) 75°

18. ABCD is a cyclic trapezium in which sides AD and BC are parallel. If ∠ABC = 85°, then what is the measure of ∠BCD?

(a) 85° (b) 95° (c) 90° (d) 80°

19. In the cyclic quadrilateral ABCD, if \(\angle\)DBA = 50° and \(\angle\)ADB = 33°, then what is the value of \(\angle\)BCD?

20. PQRS is a cyclic quadrilateral. PQ is the diameter of the circle, and if \(\angle\)PRS = 56°, then what is the value of \(\angle\)QPS?

21. If a right-angled quadrilateral has \(x\) number of vertices, \(y\) number of edges, and \(z\) number of faces, then what is the value of \(x - y + z\)?

(a) 8 (b) 6 (c) 2 (d) 12

22. If a right-angled quadrilateral has \(a\) number of vertices, \(b\) number of edges, and \(c\) number of faces, then what is the value of \(2a - b + 3c\)?

(a) 16 (b) 18 (c) 20 (d) 22

23. If ABCD is a cyclic parallelogram, what is the measure of ∠B?

(a) 90° (b) 60° (c) 45° (d) 50°

24. If the measure of an angle in degrees is \(x\) and in radians is \(y\), then what is the value of \( \frac{x}{y} \)?

(a) \(\cfrac{π}{180}\) (b) \(\cfrac{π}{90}\) (c) \(\cfrac{90}{π}\) (d) \(\cfrac{180}{π}\)

25. If two angles of a triangle are 75° and \( \frac{\pi^c}{6} \), then what is the measure of the third angle?

(a) 75° (b) 60° (c) 65° (d) 70°

26. In triangle ABC, the circumcenter is O; points A and B, C lie on opposite sides of the center. If \(\angle BOC = 120^\circ\), then what is the measure of \(\angle BAC\)?

(a) 50° (b) 60° (c) 70° (d) 80°

27. O is the circumcenter of triangle ABC. If \(\angle\)BAC = 85° and \(\angle\)BCA = 70°, then what is the measure of \(\angle\)OAC?

(a) \(65^o\) (b) \(42\cfrac{1}{2}^o\) (c) \(50^o\) (d) \(25^o\)

28. The bisectors of \(\angle\)A and \(\angle\)B of parallelogram ABCD intersect at point O. What is the measure of \(\angle\)AOB?

(a) 30° (b) 60° (c) 90° (d) 45°

29. In a right-angled quadrilateral, if the number of vertices is denoted by \(x\), the number of sides by \(y\), and the number of diagonals by \(z\), then what is the value of \(x + 3y - 5z\)?

(a) 14 (b) 44 (c) 20 (d) 24

30. Two tangents are drawn to a circle from points A and B on the circumference, and they intersect at point C. Another point P lies on the circumference, on the side opposite to where point C is located with respect to the center. If \(\angle\)APB = 35°, then what is the measure of \(\angle\)ACB?

(a) 145° (b) 55° (c) 110° (d) None of the above