1. Prove that a cyclic trapezium is an isosceles trapezium.

2. Prove that a cyclic trapezium is an isosceles trapezium and that its diagonals are equal in length.

3. Prove that a cyclic trapezium is an isosceles trapezium and that its diagonals are equal in length.

4. Let us prove that a cyclic trapezium is an isosceles trapezium and its diagonals are equal in length.

5. PQRS is a cyclic trapezium. PQ is a diameter of the circle, and PO || SR. If \(\angle\)QRS = 110°, then the value of \(\angle\)QSR is -

(a) 20° (b) 25° (c) 30° (d) 40°

6. ABCD is a cyclic trapezium. If AB || CD, then what is the relationship between AD and BC?

(a) AD = 2BC (b) AD = BC (c) AD = \(\cfrac{1}{2}\)BC (d) 3AD = 2BC

7. In the isosceles triangle ABC, AB = AC. A straight line parallel to BC intersects AB and AC at points P and Q respectively. Prove that the quadrilateral BCQP is cyclic (i.e., its vertices lie on a circle).

8. A cyclic trapezium is always an _____ trapezium.

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

10. ABCD is a cyclic trapezium, where AD \(\parallel\) BC. If \(\angle\)ABC = 70°, then the measure of \(\angle\)BCD will be—?

(a) 110° (b) 80° (c) 70° (d) 120°

11. ABCD is a cyclic quadrilateral. The extended sides AB and DC intersect at point P. Prove that: \[ PA \cdot PB = PC \cdot PD \]

12. Prove that the quadrilateral formed by the intersection of the angle bisectors of the four angles of any quadrilateral is a cyclic quadrilateral.

13. ABCD is a trapezium where AB \(\parallel\) CD. The diagonals AC and BD intersect at point O. Prove that: AO × OD = BO × OC.

14. In trapezium ABCD, AB and DC are parallel. A straight line is drawn parallel to AB, which intersects AD and BC at points E and F respectively. Prove that: \[ AE : ED = BF : FC \]

15. Prove that if the opposite angles of a quadrilateral are supplementary, then the vertices of the quadrilateral lie on a circle (i.e., the quadrilateral is cyclic).

16. 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°

17. 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°

18. A straight line intersects \( \triangle PQR \) at points \( X \) and \( Y \) on \( PQ \) and \( PR \), respectively, such that \( \frac{PX}{XQ} = \frac{PY}{YR} \). If \( \angle PXY = \angle PRQ \), then prove that \( \triangle PQR \) is an isosceles triangle.

19. 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°

20. ABCD is a cyclic quadrilateral. The bisectors of \(\angle\)DAB and \(\angle\)BCD intersect the circle at points X and Y. Prove that XY is the diameter of the circle.

21. O is the center of the circle, and AB is its diameter. ABCD is a cyclic quadrilateral. Given that \(\angle\)ABC = 65° and \(\angle\)DAC = 60°, find the measure of \(\angle\)BCD.

22. If the cyclic quadrilateral ABCD is extended along sides AB and DC, they intersect at point P. Prove that \( PA \cdot PB = PC \cdot PD \).

23. ABCD is a cyclic quadrilateral. If the extended sides AB and DC intersect at point P, prove that PA.PB = PC.PD.

24. ABCD is a cyclic quadrilateral. The extended lines AB and CD intersect at point P. Prove that PA × PB = PC × PD.

25. Prove that a cyclic parallelogram is a rectangle.

26. Prove that a cyclic parallelogram is a rectangle.

27. ABCD is a cyclic quadrilateral. The extended sides AB and DC intersect each other at point P. Prove that \[ PA \cdot PB = PC \cdot PD. \]

28. ABCD is a cyclic quadrilateral. Chord DE is the external bisector of \(\angle BDC\). Prove that AE (or the extended AE) is the external bisector of \(\angle BAC\).

29. Let us prove that if a circle is drawn using either of the equal sides of an isosceles triangle as its diameter, it bisects the unequal side of the triangle.

30. Let us prove that a cyclic parallelogram is a rectangle.