1. ABCD is a cyclic quadrilateral. If BC is the diameter and ∠ADC = 130°, then what is the value of ∠ACB?
2. ABCD is a cyclic quadrilateral. The sides AB and DC are extended to meet at point P, and the sides AD and BC are extended to meet at point Q. If ∠ADC = 85° and ∠BPC = 40°, then find the measures of ∠BAD and ∠CQD.
3. ABCD is a cyclic quadrilateral. The side BA is extended up to point F. AE is drawn parallel to CD, and ∠ABC = 92°, ∠FAE = 20°. Find the measure of ∠BCD.
4. In the cyclic quadrilateral ABCD, side AB is extended up to point P. If ∠CAB = 35° and ∠CBP = 82°, then what is the measure of ∠ADB?
(a) 47° (b) 55° (c) 35° (d) 60°
5. ABCD is a cyclic trapezium in which sides AD and BC are parallel. If ∠ABC = 85°, then what is the measure of ∠BCD?
(a) 85° (b) 95° (c) 90° (d) 80°
6. ABCD is a cyclic quadrilateral. The side AB is extended up to point X, and it is measured that ∠XBC = 82° and ∠ADB = 47°; calculate and write the value of ∠BAC.
7. 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°
8. 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°
9. In the adjacent figure, ABCD is a cyclic quadrilateral. BA is extended to point F. AE ∥ CD, ∠ABC = 92°, and ∠FAE = 20°. Find the measure of ∠BCD.
(a) 20° (b) 88° (c) 108° (d) 72°
10. If ABCD is a cyclic quadrilateral and ∠A = 100°, then the measure of ∠C is—?
(a) 50° (b) 200° (c) 80° (d) 180°
11. "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
12. If ABCD is a cyclic parallelogram, what is the measure of ∠B?
(a) 90° (b) 60° (c) 45° (d) 50°
13. 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°
14. In the cyclic quadrilateral ABCD, side AB is extended to point X. If ∠XBC = 82° and ∠ADB = 47°, then find the measure of ∠BAC.
15. 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?
16. 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?
17. 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?
18. 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°
19. In the given figure, O is the center of the circle and AB is the diameter. ABCD is a cyclic quadrilateral where AB || DC and ∠BAC = 25°. Find the measure of ∠DAC.
20. ```html AB is the diameter of a circle centered at point O. ABCD is a cyclic quadrilateral. If ∠ADC = 120°, find the value of ∠BAC. ```
(a) \(50^\circ\) (b) \(60^\circ\) (c) \(30^\circ\) (d) \(40^\circ\)
21. A circle has a radius of 2.5 cm. Two chords AB = 4 cm and AC = 3 cm are drawn. What is the measure of angle ∠BAC?
22. If ABCD is a cyclic quadrilateral and \(\angle\)A=120°, what is the measure of \(\angle\)C ?
(a) \(\cfrac{π}{3}\) (b) \(\cfrac{π}{6}\) (c) \(\cfrac{π}{2}\) (d) \(\cfrac{2π}{3}\)
23. 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—?
24. In the cyclic quadrilateral ABCD, if AB = AD, \(\angle\)DAC = 70° and \(\angle\)BDC = 50°, then what is the measure of \(\angle\)ACD?
(a) 30\(^o\) (b) 40\(^o\) (c) 50\(^o\) (d) 70\(^o\)
25. In the rhombus PQRS, if ∠Q = 60° and RS = 6 cm, what is the length of the diagonal PR?
(a) \(6\sqrt3\) cm (b) \(\sqrt3\) cm (c) \(\cfrac{1}{\sqrt3}\) cm (d) 6 cm
26. ABCD is a cyclic quadrilateral, and CD is extended up to point E. If \(\angle\)ADE = 92°, then what is the measure of \(\angle\)ABC?
(a) 88\(^o\) (b) 29\(^o\) (c) 92\(^o\) (d) 60\(^o\)
27. In the cyclic quadrilateral PQRS, the side PS is a diameter of the circle. If \(\angle\)PQR = 120°, then what is the measure of \(\angle\)SPR?
(a) 90° (b) 30° (c) 60° (d) 120°
28. In the cyclic quadrilateral PORS, the side PS is a diameter of the circle. If \(\angle\)PQR = 128°, then what is the measure of \(\angle\)SPR?
(a) 30° (b) 38° (c) 60° (d) None of the above
29. 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
30. PQRS is a cyclic quadrilateral in which side QR is extended up to point T. If the measures of angles ∠SRQ and ∠SRT are in the ratio 4:5, then find the measures of ∠SPQ and ∠SRQ.
