1. 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°

2. 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.

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. AB is a chord of a circle centered at O, and PT is a tangent to the circle at point A. If ∠ AOB = 120°, then what is the measure of ∠ BAT?

(a) 60° (b) 30° (c) 90° (d) 45°

5. In a circle with center O, PQ and PR are two chords. Tangents drawn at points Q and P intersect at point S. If ∠QSR = 70°, then what is the measure of ∠QPR?

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 extension of BA at point T. If ∠PBO = 30°, then find the measure of ∠PTA.

7. 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

8. AB is a diameter of a circle with center O. CD is a chord whose length is equal to the radius of the circle. When AC and BD are extended, they intersect at point P. What is the measure of ∠APB?

9. If two tangents are drawn from points P and Q on a circle and intersect at point A such that ∠PAQ = 60°, then what is the measure of ∠APQ?

10. 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 \(\angle PTA\)?

11. 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.

12. 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.

13. 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.

14. In the given figure, O is the center of the circle and AB is the diameter. ABCD is a cyclic quadrilateral. If ∠ADC = 120°, then find the measure of ∠BAC.

(a) 50° (b) 60° (c) 30° (d) 40°

15. In the circle with center \( O \), if \( AOB \) is a diameter and point \( C \) is on the circle such that \( AC = 3 \) cm and \( BC = 4 \) cm, what is the length of \( AB \)?

(a) 3 cm (b) 4 cm (c) 5 cm (d) 8 cm

16. 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

17. In triangle ABC, the incenter is I. When the internal bisector of ∠A (i.e., AI) is extended, it intersects the circumcircle at point P. If PB = 15 cm, then what is the length of PI?

(a) 5 cm (b) 15 cm (c) 10 cm (d) 20 cm

18. AB is a chord of a circle with center O. A tangent is drawn at point B, which intersects the extended line AO at point T. If ∠BAT = 21°, then find the value of ∠BTA.

19. AOB is a diameter of a circle. C is a point on the circle. If ∠OBC = 60°, then find the value of ∠OCA.

20. Given that ∆ABC ~ ∆DEF and the corresponding sides AB, BC, and CA of ∆ABC match with DE, EF, and DF of ∆DEF respectively; If ∠A = 47° and ∠E = 83°, then what is the measure of ∠C?

21. In triangle ABC, if O is the orthocenter and ∠BOC = 120°, then what is the measure of ∠BAC?

(a) 80° (b) 60° (c) 90° (d) 75°

22. 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 _____.

23. 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°

24. 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°

25. AB is a diameter of a circle with center O. P is any point on the circle. Tangents are drawn at points A and B, which are intersected by the tangent drawn from point P at points Q and R respectively. If the radius of the circle is \(r\), prove that \(PQ \cdot PR = r^2\).

26. Masum has drawn a circle centered at point O, in which AB is a chord. A tangent is drawn at point B, which intersects the extended line AO at point T. If ∠BAT = 21°, then calculate the measure of ∠BTA.

27. In the given figure, O is the center of the circle and AB is the diameter. ABCD is a cyclic quadrilateral. If ∠ABC = 65° and ∠DAC = 40°, then find the measure of ∠BCD.

(a) 75° (b) 105° (c) 115° (d) 80°

28. In the adjacent figure, two circles with centers P and Q intersect at points B and C. ACD is a straight line segment. If ∠ARB = 150° and ∠BQD = x°, then find the value of x.

29. “In the adjacent figure, O is the center of the circle and AC is the diameter. If ∠AOB = 80° and ∠ACE = 10°, then find the measure of ∠BED.”

30. “In the adjacent figure, O is the center of the circle and AB is the diameter. If ∠AOD = 140° and ∠CAB = 50°, then find the measure of ∠BED.”