1. 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°

2. AOB is a diameter of the circle with center O, and C is a point on the circle. If \(\angle\)OBC = 60°, then find the measure of \(\angle\)OCA.

3. O is the center of a circle. PQ is a diameter, and R is a point on the circumference. If \(\angle\)PQR = 40°, then what is the measure of \(\angle\)POR?

(a) 80° (b) 40° (c) 20° (d) 100°

4. AT is a tangent at point A on a circle with center O. BC, the extended diameter, intersects the tangent at point T. Given that \(\angle\)ABC = 25°, determine the value of \(\angle\)ATB.

5. In a circle centered at O, AB is a diameter, and C is any point on the circle. Given that \(\angle\)OAC = 45°, determine the measure of \(\angle\)OCB.

(a) 90° (b) 45° (c) 30° (d) 60°

6. 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

7. 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

8. AOB is a diameter of a circle. C is a point on the circle. If \(\angle\)OBC = 55°, find the value of \(\angle\)OCA.

9. AB is the diameter of a circle centered at O. C is any point on the circle. Given ∠BAC = 50° and CD is perpendicular to AB, Find the value of ∠BCD.

10. 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.

11. In the circle centered at O, AB is the diameter. M is a point on the circumference. If \(\angle\) MAB = 72°, find the value of \(\angle\) MBA.

(a) 72° (b) 18° (c) 108° (d) none of the above

12. ABCD is a cyclic quadrilateral with O as the center of the circle. Side DC is extended up to point P. If \(\angle BCP = 108^\circ\), calculate and write the measure of \(\angle BOD\).

13. In triangle ABC, point O is located inside the triangle such that OA = OB and \(\angle\)AOB = 2\(\angle\)ACB. If a circle is drawn with center O and radius equal to OA, then point C will lie on the circle.

14. In the adjacent figure, O is the center of the circle. Given: \(\angle\)OAB = 40°, \(\angle\)ABC = 120°, \(\angle\)BCO = y°, and \(\angle\)COA = x° Find the values of x and y.

15. In the adjacent figure, O is the center of the circle. Given \(\angle\)BAD = 65° and \(\angle\)BDC = 45°, find the value of \(\angle\)CBD.

(a) 65° (b) 45° (c) 40° (d) 20°

16. 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°

17. 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°

18. 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.

19. In the adjacent figure, AB is the diameter of a circle centered at O. Point C is any point on the circle. Given that ∠BAC = 50° and CD is perpendicular to AB, find the value of ∠BCD.

20. 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 \(\angle ACB\).

21. QR is a chord of a circle with center O. Tangents are drawn at points Q and R, and they intersect each other at point P. QM is the diameter of the circle. Prove that \(\angle\)QPR = 2\(\angle\)RQM.

22. In a circle with center \(O\), \(AB\) is a diameter. \(P\) is any point on the circumference. If \(\angle POA = 120°\), then the measurement of \(\angle PBO\) is –

(a) 30° (b) 60° (c) 90° (d) 120°

23. 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

24. In the adjacent figure, O is the center of the circle and BOA is the diameter. A tangent is drawn at point P on the circle, which intersects the extended line BA at point T. If \(\angle PBO = 30^\circ\), then what is the measure of \(\angle PTA\)?

25. AB is a diameter of a circle centered at O. P is any point on the circumference. If \(\angle\)POA = 120°, then find the measure of \(\angle\)PBO.

26. ABCD is a cyclic quadrilateral where AB \(\parallel\) DC, and O is the center of the circle. Given: \(\angle\)BOC = 80°, and \(\angle\)ACO = 10° Find the measure of \(\angle\)BAD.

27. Given: In a circle with center O, MN is any chord that is not a diameter, and AC is a diameter. To Prove: AC > MN, i.e., the diameter is the longest chord of a circle. Construction: From the center O, draw a perpendicular OD to the chord MN. Join O and M. Proof: OM > MD [∵ Triangle OMD is a right-angled triangle and OM is the hypotenuse] Or, OA > MD [∵ OA = OM, both are radii of the same circle] Or, ½AC > ½MN Or, AC > MN Therefore, the diameter is the longest chord of a circle.

28. 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.

29. In a circle centered at O, AB is a diameter. ABCD is a cyclic quadrilateral. Given \(\angle\)ABC = 65° and \(\angle\)DAC = 40°, find the value of \(\angle\)BCD.

30. AB is a chord of the circle centered at O. PT is the tangent at point B. If \(\angle\) ABT = 54°, then find the value of \(\angle\) AOB.