1. 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.

2. 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°

3. "AB is a diameter of a circle with center O. P is any point on the circumference. If ∠ POA = 120°, then what is the measure of ∠ PBO?"

(a) 90\(^o\) (b) 60\(^o\) (c) 75\(^o\) (d) None of the above

4. 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

5. 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°

6. 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 \(\angle PTA\)?

7. AB is a diameter. If P is any point on the circle, then the value of \(\angle\)APB is –

(a) 90° (b) 180° (c) 45° (d) None of the above

8. 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

9. 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.

10. 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°

11. 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.

12. 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°

13. In a circle centered at \(O\), \(AB\) is a diameter. \(ABCD\) is a cyclic quadrilateral. If \(\angle ADC = 120^\circ\), then the measure of \(\angle BAC\) is –?

(a) 60° (b) 40° (c) 50° (d) 30°

14. In a circle centered at O, PQ is a diameter; R is a point on the circle, and PR = RQ. Then, the measure of \(\angle\)RPQ is: ____?

(a) 30° (b) 90° (c) 60° (d) 45°

15. 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.

16. From an external point \(P\), two tangents \(PS\) and \(PT\) are drawn to a circle with center \(O\). \(QS\) is a chord of the circle that is parallel to \(PT\). If \(\angle SPT = 80^\circ\), then what is the measure of \(\angle QST\)?

17. In the center circle O, AB is a diameter. C is any point on the circumference of the circle where AC = 3 cm and BC = 4 cm. Find the length of AB.

(a) 3 cm (b) 4 cm (c) 5 cm (d) 7 cm

18. Two circles touch each other externally at point \(C\). \(AB\) is a common tangent to the two circles, touching them at points \(A\) and \(B\), respectively. The measurement of \(\angle ACB\) is –

(a) 60° (b) 45° (c) 30° (d) 90°

19. 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°

20. 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°

21. AB and AC are two tangents drawn from point A to a circle with center O. The line OA intersects the chord BC (which joins the points of contact) at point M. If AM = 8 cm and BC = 12 cm, then what is the length of each tangent?

(a) 8 cm (b) 10 cm (c) 12 cm (d) 16 cm

22. 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.

23. 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.

24. Two tangents are drawn to a circle from points A and B on its circumference, and they intersect at point P. If \(\angle\)APB = 68°, then what is the measure of \(\angle\)PAB?

25. In a circle with center O, AB and CD are two equal chords. If \(\angle\)AOB = 60° and CD = 10 cm, then what is the length of OD?

26. AB and CD are two chords of a circle with center O, and both have equal lengths. If \(\angle\)AOB = 60°, then what is the measure of \(\angle\)COD?

27. AOB is the diameter of a circle centered at O. C is a point on the circumference. If AC = 3 cm and BC = 4 cm, then the length of AB will be _____.

28. 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.

29. In the given figure, \(O\) is the center of the circle, and \(AB\) is the diameter. If \(\angle ADC = 120^\circ\), then determine the value of \(\angle BAC\).

(a) 50° (b) 60° (c) 40° (d) 30°

30. 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 \(\angle BOD\).