1. In a circle with center \(O\), \(AB\) is the diameter, and \(P\) is a point on the circle. If \(\angle AOP = 104°\), find the value of \(\angle BPO\).

(a) 54° (b) 72° (c) 36° (d) 27°

2. Two chords, \(AB\) and \(CD\), of a circle with center \(O\), intersect at point \(P\). If \(\angle APC = 40°\), find the value of \(\angle AOC + \angle BOD\).

(a) 60° (b) 80° (c) 120° (d) None of these

3. 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 \(\angle BAC\).

4. In the cyclic quadrilateral ABCD, if AB = AD and \(\angle\)ABD = 30°, then find the value of \(\angle\)BCD.

(a) 90° (b) 30° (c) 45° (d) 60°

5. If the cyclic quadrilateral ABCD has its sides AB and DC extended to meet at point P, and sides AD and BC extended to meet at point Q, then given \(\angle\)ADC = 85° and \(\angle\)BPC = 40°, find the value of \(\angle\)CQD.

6. AOB is the diameter of a circle. AC and BD are two chords that intersect at point E. If \(\angle\)COD = 40°, find the value of \(\angle\)CED.

7. ABCD is a cyclic quadrilateral. If \(\angle\)ABC = 65° and \(\angle\)CAD = 40°, then find the value of \(\angle\)BCD.

(a) 70° (b) 25° (c) 115° (d) 90°

8. ABCD is a cyclic quadrilateral. If \(\angle\) BAD = 65°, \(\angle\) ABD = 70°, and \(\angle\) BDC = 45°, then find the value of \(\angle\) ACB.

9. In the adjacent figure, O is the center of the circle. If \(\angle\)AEB = 110° and \(\angle\)CBE = 30°, find the value of \(\angle\)ADB.

(a) 70° (b) 60° (c) 80° (d) 90°

10. In the adjacent figure, O is the center of the circle. If \(\angle\)BCD = 28° and \(\angle\)AEC = 38°, find the value of \(\angle\)AXB.

(a) 56° (b) 86° (c) 38° (d) 28°

11. In the adjacent figure, O is the center of the circle and AB is the diameter. AB ∥ CD. If \(\angle\)ABC = 25°, find the value of \(\angle\)CED.

(a) 80° (b) 50° (c) 25° (d) 40°

12. In the adjacent figure, O is the center of the circle, AC is the diameter, and chord DE is parallel to diameter AC. If \(\angle\)CBD = 60°, find the value of \(\angle\)CDE.

13. In the adjacent figure, QS is the bisector of \(\angle\)PQR. If \(\angle\)SQR = 35° and \(\angle\)PRQ = 32°, find the value of \(\angle\)QSR.

14. In the adjacent figure, O is the center of the circle and AB = AC. If \(\angle\)ABC = 32°, find the value of \(\angle\)BDC.

15. In triangles ABC and DEF, \(\angle A = \angle E = 40^\circ\), AB : ED = AC : EF, and \(\angle F = 65^\circ\). Find the measure of \(\angle B\).

(a) 35° (b) 65° (c) 75° (d) 85°

16. 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 \(\angle ACB\).

17. Two circles intersect at points A and B. Two straight lines passing through points A and B intersect the circles at points P, Q and R, S respectively. If \(\angle\)APR = 80° and \(\angle\)BRP = 78°, then find the values of \(\angle\)AQS and \(\angle\)BSQ.

18. In a circle centered at O, the chords AB and CD have equal lengths. If \(\angle\)AOB = 60°, then the value of \(\angle\)COD is -

(a) 60° (b) 30° (c) 120° (d) 90°

19. 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°

20. In a circle with center \(O\), \(\bar{AB}\) is a diameter. On the opposite side of the circumference from the diameter \(\bar{AB}\), there are two points \(C\) and \(D\) such that \(\angle AOC = 130°\) and \(\angle BDC = x°\). Find the value of \(x\).

(a) 25° (b) 50° (c) 60° (d) 65°

21. In triangle ABC, the circumcenter is O. It is given that \(\angle\)BAC = 85° and \(\angle\)BCA = 55°. Find the value of \(\angle\)OAC.

22. ABCD is a cyclic quadrilateral and O is the center of the circle. Given: \(\angle COD = 130^\circ\), \(\angle BAC = 25^\circ\) Find the values of \(\angle BOC\) and \(\angle BCD\).

(a) 40\(^o\),90\(^o\) (b) 50\(^o\),90\(^o\) (c) 65\(^o\),50\(^o\) (d) None of the above

23. In triangle ABC, O is the circumcenter and D is the midpoint of side BC. If \(\angle\)BAC = 40°, find the measure of \(\angle\)BOD.

24. In a circle with center O, two chords AB and CD intersect at point P. If \(\angle\)AOD = 100° and \(\angle\)BOC = 70°, find the measure of \(\angle\)APC.

25. In a circle with center O, triangle ABC is inscribed. If \(\angle\)BAC = 85° and \(\angle\)BCA = 75°, then find the measure of \(\angle\)AOC.

26. In \(\triangle\)ABC, \(\angle\)ABC = 90° and BD \(\perp\) AC. If BD = 6 cm and AD = 4 cm, then find the length of CD.

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

28. 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?

29. In a circle centered at O, the chords AB and CD are of equal length. If \(\angle\)AOB = 60°, then determine the value of \(\angle\)COD.

(a) 40° (b) 30° (c) 60° (d) 90°

30. ABCD is a cyclic quadrilateral. The sides AD and AB are extended to E and F, respectively. If \(\angle\) CBF = 120°, then find the measure of \(\angle\) CDE.