1. Prove that all angles subtended by the same arc of a circle are equal.

2. The measure of all inscribed angles subtended by the same arc in a circle is equal.

3. Prove that the angle subtended by an arc at the center of a circle is twice the angle subtended by the same arc at any point on the circumference.

4. Prove that the angle subtended by an arc at the center of a circle is double the angle subtended by the same arc at any point on the remaining part of the circumference.

5. If the central angle of a circle centered at point O is 300°, then the angle subtended at the circumference by the same arc will be 60°.

6. Prove that the central angle subtended by an arc of a circle is twice any inscribed angle subtended by the same arc.

7. If two angles subtended by two arcs of the same circle are equal, then the lengths of the arcs are —.

8. Prove that the central angle subtended by an arc of a circle is twice the inscribed angle subtended by the same arc.

9. Prove that the central angle subtended by an arc of a circle is twice the inscribed angle subtended by the same arc.

10. Prove that the central angle subtended by an arc of a circle is twice any inscribed angle subtended by the same arc.

11. Prove that the central angle subtended by an arc of a circle is twice any inscribed angle subtended by the same arc.

12. Prove that the angle subtended at the center by an arc of a circle is twice the angle subtended by the same arc at any point on the circle.

13. Prove that the angle subtended at the center of any circle by an arc is double the angle subtended by the same arc at any point on the remaining part of the circle.

14. If the angles subtended by two arcs of the same circle are equal, then the lengths of the arcs are --------.

15. "Chords PQ and RS of a circle intersect each other at point X inside the circle. By joining P to S and R to Q, prove that triangles ∆PXS and ∆RSQ are similar. From this, prove that: PX × XQ = RX × XS Or, when two chords of a circle intersect internally, the product of the two segments of one chord is equal to the product of the two segments of the other chord.

16. The numerical values of the circumference and the area of a circular region are equal. The length of the diagonal of the square circumscribed about that circle is —

(a) 4 units (b) 2units (c) \(4\sqrt{2}\) units (d) \(2\sqrt{2}\) units

17. An inscribed angle subtended by the same arc is half of the central angle subtended by that arc.

18. The radius of a circle is 28 cm. What is the circular measure of the central angle subtended by an arc of length 5.5 cm in that circle?

19. If the angle formed between two intersecting chords of a circle is bisected by a line that passes through the center, then prove that the two chords are equal.

20. If the circle drawn by Shubho has a radius of 7 cm, then calculate and write the radian measure of the central angle subtended by an arc of length 5.5 cm on that circle.

21. Let ∆ABC be inscribed in a circle. AD and AE are diameters of the circle and perpendicular to side BC, intersecting BC at point E. To prove: ∆AEB and ∆ACD are similar Proof: ∠AEB = ∠ACD (both are right angles) ∠BAE = ∠DAC (angles in the same segment) So, ∆AEB ∼ ∆ACD (by AA similarity) Now, since the triangles are similar: \[ \frac{AB}{AE} = \frac{AC}{AD} \] Cross-multiplying: \[ AB \cdot AD = AC \cdot AE \] So, \[ AB \cdot AC = AE \cdot AD \quad \text{(proved)} \]

22. Prove that, in an isosceles trapezium, the two angles adjacent to either of the parallel sides are equal.

23. Triangles \(∆ABC\) and \(∆DBC\) are drawn on the same base \(BC\) and on the same side of it. Let \(E\) be any point on side \(BC\). From point \(E\), lines parallel to \(AB\) and \(BD\) are drawn, intersecting sides \(AC\) and \(DC\) at points \(F\) and \(G\) respectively. Prove that: \(AD ∥ FG\).

24. Mohit has drawn two straight lines from an external point of a circle, which intersect the circle at points A, B and C, D respectively. Using reasoning, prove that two angles of ∆XAC and ∆XBD are equal.

25. 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°

26. After a spherical ice ball starts melting, it is observed that its curved surface area has reduced by 75% after 15 minutes. The percentage of ice that has melted during this time is –

(a) 75 (b) 87.5 (c) 50 (d) 92.5

27. The radius of a circle is 28 cm. What is the radian measure of the central angle subtended by an arc of length 5.5 cm on this circle?

28. Prove that the central angle subtended by the same arc is twice the inscribed angle.

29. Two friends start a partnership business by investing ₹40,000 and ₹50,000 respectively. They agree that 50% of the profit will be shared equally, and the remaining 50% will be divided in the ratio of their investments. If the first friend’s share of the profit is ₹800 less than the second friend’s share, then what is the first friend’s profit?

30. If AB and AC are chords of the larger of two concentric circles, and they touch the smaller circle at points P and Q respectively, prove that: \[ PQ = \frac{1}{2}BC \]