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. In a circle centered at \(O\), two chords \(AB\) and \(CD\) have equal lengths. Given that \(\angle AOB = 60°\) and the radius of the circle is \(6\) cm, find the area of \(\triangle COD\).

(a) \(9\sqrt3\) square c.m (b) \(6\sqrt3\) square c.m (c) \(2\sqrt3\) square c.m (d) \(3\sqrt3\) square c.m

3. 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?

4. In a circle centered at \(O\), two equal chords \(AB\) and \(CD\) are given. If \(\angle AOB = 60^\circ\) and the radius of the circle is 6 cm, determine the area of \( \triangle COD \).

(a) \(6\sqrt3\) square cm (b) \(2\sqrt3\) square cm (c) \(2\sqrt3\) square cm (d) \(9\sqrt3\) square cm

5. 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 \(\angle QST\)?

6. 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°

7. 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

8. 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?

9. From an external point \(P\), two tangents \(PA\) and \(PB\) are drawn to a circle centered at \(O\). Given \(PA = 9\) cm and \(\angle APB = 60°\), find the length of \(AB\).

10. 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 \(\angle ACB\).

11. 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

12. 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°

13. 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 \(\angle BOD\).

14. ABCD is a cyclic quadrilateral and O is the center of the circle. If \(\angle COD = 120^\circ\) and \(\angle BAC = 30^\circ\), then calculate the values of \(\angle BOC\) and \(\angle BCD\).

15. We have drawn two circles with centers \(A\) and \(B\), which touch each other externally at point \(C\). A point \(O\) lies on the common tangent at point \(C\), and tangents \(OD\) and \(OE\) are drawn from point \(O\) to the circles centered at \(A\) and \(B\), touching them at points \(D\) and \(E\) respectively. It is given: - \(\angle COD = 56^\circ\) - \(\angle COE = 40^\circ\) - \(\angle ACD = x^\circ\) - \(\angle BCE = y^\circ\) We are to prove: - \(OD = OC = OE\) - \(x - y = 4^\circ\)

16. 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°

17. Here’s the English translation of your math problem: In triangle \( \triangle ABC \), \( \angle ABC = 90^\circ \), \( BC = 24 \) cm, and \( E \) is the midpoint of \( AC \). If \( ED \perp BC \), then what is the length of \( BD \)?

(a) 6 cm (b) 8 cm (c) 9 cm (d) None of the above

18. 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

19. In a circle centered at O, ABCD is a cyclic quadrilateral. Given: \(\angle ABD = 50^\circ\), \(\angle CAD = 28^\circ\), and \(\angle ADB = 32^\circ\) Find the measure of \(\angle BCD\).

(a) 72° (b) 52° (c) 62° (d) 82°

20. 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°

21. If \( O \) is the circumcenter of triangle \( ABC \) and \( \angle BAC = 40^\circ \), determine the value of \( \angle OBC \).

(a) 100° (b) 30° (c) 50° (d) 40°

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

(a) 30° (b) 40° (c) 120° (d) 60°

23. In a circle centered at O, PQ and PR are two chords. The tangents drawn at points Q and R intersect at point S. If \(\angle\)QSR = 70°, determine the value of \(\angle\)QPR.

24. AB and CD are two chords of a circle with center O. When extended, they intersect at point P. Prove that \(\angle AOC - \angle BOD = 2\angle BPC\).

25. In the adjacent figure, O is the center of the circle, \(\angle AOD = 40^\circ\), and \(\angle ACB = 35^\circ\). Calculate and write the measures of \(\angle BCO\) and \(\angle BOD\), and provide reasoning to support your answers.

26. In the adjacent figure, O is the center of the circle and \(\angle APB = 80^\circ\). Determine the sum of the measures of \(\angle AOB\) and \(\angle COD\), and provide reasoning to support your answer.

27. In a circle with center O, chords AB and AC are equal in length. If \(\angle\)APB and \(\angle\)DQC are inscribed angles, then the measures of the two angles are _____________.

28. O is the circumcenter of triangle ABC. If \(\angle BAC = 85^\circ\) and \(\angle BCA = 55^\circ\), then determine the value of \(\angle OAC\).

29. In triangle \(ABC\) and triangle \(DEF\), \(\angle ABC = \angle DEF\), \(\frac{AB}{DE} = \frac{BC}{EF}\), and the ratio of their areas is \(\frac{\triangle ABC}{\triangle DEF} = \frac{25}{16}\). If \(BC = 10\) cm, what is the length of \(EF\)?

30. 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°