1. 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 _____________.
2. Two chords AB and CD of a circle with center O intersect each other at point P. Prove that \(\angle AOD + \angle BOC = 2\angle BPC\) If \(\angle AOD\) and \(\angle BOC\) are supplementary, then prove that the two chords are perpendicular to each other.
3. If \(\angle\)A = 65° in parallelogram ABCD, then the measures of \(\angle\)B, \(\angle\)C, and \(\angle\)D respectively are:
(a) 65°, 115°, 115° (b) 115°, 115°, 65° (c) 115°, 65°, 115° (d) 65°, 65°, 115°
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. In a circle, AB and AC are two chords. Tangents are drawn at points B and C, and they intersect at point P. If \(\angle\)BAC = 54°, then what is the measure of \(\angle\)BPC?
6. ABC is a right-angled triangle with \(\angle ABC = 90^\circ\); if the coordinates of points A and C are (0, 4) and (3, 0) respectively, then the area of triangle ABC is:
(a) 12 square units (b) 6 square units (c) 24 square units (d) 8 square units
7. ABCD is a cyclic trapezium in which sides AD and BC are parallel to each other. If \(\angle\)ABC = 75°, then what is the measure of \(\angle\)BCD?
(a) 105° (b) 90° (c) 150° (d) 75°
8. 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?
9. 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?
10. 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?
11. In a circle centered at O, OA and OB are two radii, and \(\angle\)AOB = 130°. If the tangents drawn at points A and B meet at point T, then \(\angle\)ATB = 25°.
12. 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°
13. ABCD is a cyclic quadrilateral. The sides AD and AB are extended to E and F, respectively. If \(\angle\) CBF = 120°, then find the measure of \(\angle\) CDE.
14. ABCD is a cyclic trapezium where AD and BC are parallel sides. If \(\angle\)ABC = 75°, then the measure of \(\angle\)BCD is –
(a) 30° (b) \(\frac{75°}{2}\) (c) 45° (d) 75°
15. The chords AB and CD of a central circle are equal in length. If \(\angle\)COD = 60°, then \(\angle\)AOB = ———.
16. If the chords AB and AC of a central circle are located on opposite sides of the radius OA, then \(\angle\)OAB = \(\angle\)OAC.
17. If \(AC = BC\) in a triangle and \(AB^2 = 2AC^2\), then the measure of \(\angle C\) is _____.
(a) 30° (b) 45° (c) 60° (d) 90°
18. If \(AC = BC\) in a triangle and \(AB^2 = 2AC^2\), then the measure of \(\angle C\) is _____.
19. In \(\triangle\)ABC, if AB = \((2a-1)\) cm, AC = \(2\sqrt{2}a\) cm, and BC = \((2a+1)\) cm, then write the value of \(\angle\)BAC.
20. In \(\triangle\)ABC, if AB \(= (2p-1)\) cm, AC \(= 2\sqrt2p\) cm, and BC \(= (2p+1)\) cm, then the value of \(\angle\)BAC is...? Let me know if you need further assistance!
21. In a circle centered at O, the chords AB and CD are equal in length. If \(\angle\)AOB = 60°, then find the measure of \(\angle\)COD.
22. In a circle centered at O, if the lengths of chords PQ and RS are in the ratio 1:1, then the ratio of \(\angle\)POQ to \(\angle\)ROS is _____ |
23. When the sides AB and DC of a cyclic quadrilateral ABCD are extended, they meet at point P; and when sides AD and BC are extended, they meet at point Q. If \(\angle\)ADC = 85° and \(\angle\)BPC = 40°, then calculate the measures of \(\angle\)BAD and \(\angle\)CQD.
24. ABCD is a cyclic quadrilateral. If \(\angle ADB = x^\circ\) and \(\angle ABD = y^\circ\), then the measure of \(\angle BCD\) will be \((x + y)^\circ\).
25. 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 \(\angle QST\)?
