1. If the cyclic quadrilateral ABCD has its sides AB and DC extended to meet at point P, and sides AD and BC extended to meet at point Q, then given \(\angle\)ADC = 85° and \(\angle\)BPC = 40°, find the value of \(\angle\)CQD.

2. 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.

3. PQRS is a cyclic quadrilateral, and the sides PQ and SR are extended to meet at point T. The center of the circle is O; if ∠POQ = 110°, ∠QOR = 60°, and ∠ROS = 80°, then calculate and write the values of ∠RQS and ∠QTR.

4. 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.

5. 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.

6. ABCD is a cyclic quadrilateral. If the sides AB and DC are extended, they meet at point P; and if the sides AD and BC are extended, they meet at point R. The circumcircles of triangles BCP and CDR intersect at point T. Prove that the points P, T, and R lie on a straight line.

7. In triangle \(\triangle ABC\), AB = AC. Points E and F are the midpoints of sides AB and AC respectively. AD is perpendicular to BC, and AD = 4 cm. If EF = 3 cm, then what is the length of BD?

(a) 4 cm (b) 3 cm (c) 6 cm (d) 7 cm

8. A straight line parallel to side BC of triangle ∆ABC intersects sides AB and AC at points D and E respectively. If AD : BD = 3 : 5, then what is the ratio of the area of triangle ∆ADE to the area of trapezium DBCE?

9. In the cyclic quadrilateral ABCD, side AB is extended to point X. If ∠XBC = 82° and ∠ADB = 47°, then find the measure of ∠BAC.

10. 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.

11. ABCD is a cyclic quadrilateral. Side AB is extended to point X. If \(\angle XBC = 82^\circ\) and \(\angle ADB = 47^\circ\), find the value of \(\angle BAC\).

12. In the cyclic quadrilateral ABCD, the sides AB and AD are extended to E and F, respectively. If \(\angle\)CBE = 120°, find the measure of \(\angle\)CDF.

13. In the adjacent figure, O is the center of the circle and AB is the diameter. The length of chord CD is equal to the radius of the circle. AC and BD are extended to meet at point P. Find the value of ∠APB.

14. The adjacent cyclic quadrilateral ABCD has its sides AD and AB extended to points E and F, respectively. If \(\angle\)CBF = 120°, then calculate the value of \(\angle\)CDE.

15. 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°