1. If ABCD is a cyclic quadrilateral and \(\angle\)A=120°, what is the measure of \(\angle\)C ? (a) \(\cfrac{π}{3}\) (b) \(\cfrac{π}{6}\) (c) \(\cfrac{π}{2}\) (d) \(\cfrac{2π}{3}\)

2. In cyclic quadrilateral ABCD, the extended sides AB and DC intersect at point P. Given: PA = 6 cm, PB = 2 cm, and PD = 8 cm Find: The length of PC. (a) 3 cm (b) 1.5 cm (c) 4.5 cm (d) 6 cm

3. ABCD is a cyclic quadrilateral. The extended lines AB and DC intersect at point P, then— (a) PA.PB=PC.PD (b) PA.PC=PPDB. (c) PA.PC\(\lt\)PB.PD (d) PA.PB\(\gt\)PC.PD

4. 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\(^o\)

5. If PQRS is a cyclic parellelogram, what is the value of \( \angle P \) ? (a) 45° (b) 60° (c) 90° (d) 75°

6. The measure of each angle of a cyclic parallelogram. (a) 90\(^o\), 90\(^o\), 100\(^o\), 80\(^o\) (b) 100\(^o\), 80\(^o\), 100\(^o\), 80\(^o\) (c) 90\(^o\), 90\(^o\), 90\(^o\), 90\(^o\) (d) none of the above

7. In the cyclic quadrilateral ABCD, if \(\angle\)A = 120°, then the measure of \(\angle\)C in a circular sense. (a) \(\cfrac{π}{2}\) (b) \(\cfrac{π}{3}\) (c) \(\cfrac{π}{6}\) (d) \(\cfrac{π}{4}\)

8. If ABCD is a cyclic parallelogram, what is the measure of ∠B? (a) 90° (b) 60° (c) 45° (d) 50°

9. PQRS is a cyclic trapezium. PQ is a diameter of the circle, and PO || SR. If \(\angle\)QRS = 110°, then the value of \(\angle\)QSR is - (a) 20° (b) 25° (c) 30° (d) 40°

10. 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\(^o\)

11. ABCD is a cyclic trapezium. If AB || CD, then what is the relationship between AD and BC? (a) AD = 2BC (b) AD = BC (c) AD = \(\cfrac{1}{2}\)BC (d) 3AD = 2BC

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. In the cyclic quadrilateral \(ABCD\), if \(\angle A = 120°\), find the measure of the angle \(\angle C\). (a) \(\cfrac{π}{3}\) (b) \(\cfrac{π}{6}\) (c) \(\cfrac{π}{2}\) (d) \(\cfrac{2π}{3}\)

14. ABCD is a cyclic quadrilateral. If ∠ ABD = 48°, what is the measure of ∠ ACD? (a) 42° (b) 138° (c) 48° (d) 12°

15. ABCD is a cyclic trapezium. If AB \(\parallel\) CD, then what is the relationship between AD and BC? (a) AD = 2BC (b) AD = BC (c) AD = \(\cfrac{1}{2}\)BC (d) 3AD = 2BC

16. 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

17. If ABCD is a cyclic quadrilateral and ∠A = 100°, then the measure of ∠C is—? (a) 50° (b) 200° (c) 80° (d) 180°

18. 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°

19. ABCD একটি বৃত্তস্থ চতুর্ভুজ। CD-কে E পর্যন্ত বর্ধিত করা হল। যদি \(\angle\)ADE = 70° হয়, তাহলে \(\angle\)ABC-এর মান হবে - translate in english (a) 140\(^o\) (b) 35\(^o\) (c) 105\(^o\) (d) 70\(^o\)

20. ABCD is a cyclic trapezium, where AD \(\parallel\) BC. If \(\angle\)ABC = 70°, then the measure of \(\angle\)BCD will be—? (a) 110° (b) 80° (c) 70° (d) 120°

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

22. Prove that if the opposite angles of a quadrilateral are supplementary, then the vertices of the quadrilateral lie on a circle (i.e., the quadrilateral is cyclic).

23. In the isosceles triangle ABC, AB = AC. A straight line parallel to BC intersects AB and AC at points P and Q respectively. Prove that the quadrilateral BCQP is cyclic (i.e., its vertices lie on a circle).

24. ABCD is a cyclic quadrilateral. The sides AB and DC are extended to meet at point P, and the sides AD and BC are extended to meet at point Q. If ∠ADC = 85° and ∠BPC = 40°, then find the measures of ∠BAD and ∠CQD.

25. In a circle with center O, where BOC is the diameter and ABCD is a cyclic quadrilateral, if \(\angle ADC = 110^\circ\), then find the value of \(\angle ACB\).

26. Prove that the opposite angles of a cyclic quadrilateral are supplementary.

27. If ABCD is a cyclic quadrilateral inscribed in a circle with center O, then prove that AB + CD = AD + BC.

28. Prove that the quadrilateral formed by the intersection of the angle bisectors of the four angles of any quadrilateral is a cyclic quadrilateral.

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

30. In the cyclic quadrilateral ABCD, the side AB is the diameter of the circle. If \(\angle ACD = 50^\circ\), find the measure of \(\angle BAD\).

31. ABCD is a cyclic quadrilateral. If \(\angle ADB = x^\circ\) and \(\angle ABD = y^\circ\), then the measure of \(\angle BCD\) will be \((x + y)^\circ\).

32. If ABCD is a cyclic parallelogram, then the measure of ∠A will be ——.

33. Prove that a cyclic trapezium is an isosceles trapezium.

34. Prove that the opposite angles of a cyclic quadrilateral are supplementary.

35. ABCD is a cyclic quadrilateral. Chord DE is the external bisector of \(\angle BDC\). Prove that AE (or the extended AE) is the external bisector of \(\angle BAC\).

36. ABCD is a cyclic quadrilateral. The side BA is extended up to point F. AE is drawn parallel to CD, and ∠ABC = 92°, ∠FAE = 20°. Find the measure of ∠BCD.

37. ABCD is a cyclic quadrilateral. The angle bisectors of \(\angle\)DAB and \(\angle\)BCD intersect the circle at points X and Y respectively. If O is the center of the circle, find the value of \(\angle\)XOY.

38. If the opposite angles of a quadrilateral are supplementary, then the vertices of the quadrilateral lie on the same circle.

39. If one side of a cyclic quadrilateral is extended, the exterior angle thus formed is equal to the interior opposite angle — prove it.

40. If \( BC || AD \), \( \angle CDE = 80^\circ \), and \( \angle CBD = 30^\circ \), determine the values of \( \angle ABD \), \( \angle ACD \), and \( \angle BAD \).

41. In the cyclic quadrilateral ABCD, side AB is extended to point X. If ∠XBC = 82° and ∠ADB = 47°, then find the measure of ∠BAC.

42. ABCD is a cyclic quadrilateral and O is the center of the circle. If ∠COD = 120° and ∠BAC = 30°, then find the values of ∠BOC and ∠BCD.

43. Prove that a parallelogram with equal diagonals must be a rectangle.

44. Prove that the opposite angles of a cyclic quadrilateral are supplementary.

45. ABCD is a cyclic quadrilateral. Side AB is extended to point X. If \(\angle XBC = 82^\circ\) and \(\angle ADB = 47^\circ\), find the value of \(\angle BAC\).

46. 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?