1In a circle with centre O, the diagonals AC and BD of ABCD cross. \( \angle BAC = 30^\circ \), \( \angle DBC = 70^\circ \) and \( x = \angle BCD \). Find x.
2AB is a diameter of a circle with centre O, and P and Q are on the circle. \( \angle PBA = 35^\circ \) and \( x = \angle PQB \). Find x.
3A, B and C lie on a circle with centre O. \( \angle OBA = 25^\circ \), \( \angle OCA = 35^\circ \) and \( x = \angle BOC \). Find x.
4In a circle with centre O, AO is extended to D. \( \angle BOD = 90^\circ \), \( \angle DOC = 60^\circ \) and \( x = \angle BAC \). Find x.
5AB and CD are diameters of a circle with centre O. \( \angle CAB = 50^\circ \) and \( x = \angle ABD \). Find x.
6A, B and C lie on a circle with centre O. The angle AOC on the side of B is \( 130^\circ \), and \( x = \angle ABC \). Find x.
7AB is a diameter, C is on the circle and CD is perpendicular to AB. \( \angle CAB = 50^\circ \) and \( x = \angle DCB \). Find x.
8AB is a diameter through the centre O. \( \angle AOC = 130^\circ \) and \( x = \angle CDB \). Find x.
9A, B, C and D are on a circle, and AC and BD meet at P. \( \angle BAC = 40^\circ \) and \( \angle DBC = 80^\circ \). Find x, the angle of triangle BCD at C.
10B and E are the ends of a diameter of a circle with centre O; A and D are on the circle. \( \angle OAE = 50^\circ \) and \( x = \angle ADB \). Find x.
11O is the centre of a circle, \( \angle OBA = 25^\circ \) and \( \angle OCA = 30^\circ \). Find the obtuse \( \angle BOC \).
12O is the centre and QR is a diameter. If \( \angle PQR = 40^\circ \) and \( \angle PSQ = x \), find x.
13In a circle with centre O, FAEC is a cyclic quadrilateral and AE is produced to D. If \( \angle CED = 68^\circ \), find \( \angle AFC \) and the angle AOC.
14AOB is a diameter of the circle. If \( \angle ADC = 100^\circ \), find \( \angle BAC \).
15B, C, D and E lie on one chord, BC = DE and arc AB = arc FE. Prove that \( \angle ACB = \angle FDE \).
16If \( \angle APC = \angle BQD \), prove that \( AB \parallel CD \).
17O is the centre and AB is a diameter. If arc PQ = arc PB, prove that \( AQ \parallel OP \).
18Chords AC and BD meet at P and PB = PC. Prove that chord AB = chord DC, chord AC = chord BD, and arc ABC = arc BCD.
19O is the centre and DG is a diameter. Chords DE and FG meet at H. Prove that \( \angle DOF + \angle EOG = 2\angle EHG \).
20Chords MN and RS of a circle meet outside it at X. Prove that \( \angle MXR \) is half the difference of arc MR and arc NS, in degree measure.
21PQRS is a cyclic quadrilateral. The bisectors of \( \angle QPS \) and \( \angle QRS \) meet the circle at A and B respectively. Prove that AB is a diameter.
22Two circles meet at A and Q; NPS, MAN and RMS are straight lines. Prove that PQRS is a cyclic quadrilateral.
23PQRS is a parallelogram; SP produced meets the circle at U and RQ meets it at T. Prove that UTRS is a cyclic quadrilateral.
24ABCD is a parallelogram and ABFE a rectangle on the same base AB. What is the relationship between their areas? If AB = BM, write the relationship between parallelogram ABCD and triangle AMN.
25A parallelogram ABCD and a rectangle ABFE are on the same base AB and between the parallels AB and EC. If ABFE is 35 cm², what is the area of ABCD? Are triangles AED and BFC congruent?
26Construct a parallelogram ABCD with AB = 7 cm, BC = 5 cm and \( \angle ABC = 120^\circ \), and a rectangle ABFE equal to it in area.
27Triangle ABC with \( \angle BAC = 90^\circ \), AB = 7 cm and BC = 25 cm, and parallelogram BCED are on the same base BC between the same parallels. Find AC and the area of BCED, and prove that the triangle is half the parallelogram.
28Construct triangle ABC with AC = 5 cm, AB = 4 cm and \( \angle BAC = 45^\circ \), and a parallelogram ADMN equal to it in area.
29ABCD and PQRD are two parallelograms. Find the relation between them. If ABCD has base 8 cm and height 7 cm, find the area of PQRD.
30X is inside parallelogram ABCD and \( MN \parallel AB \). Prove that triangles XCD and XAB together make half of ABCD. What is the relationship among triangles AXD, BCX and ABCD?
31Construct a parallelogram ABCD with AB = 5 cm, BC = 4 cm and \( \angle ABC = 60^\circ \), and a triangle PBE equal to it in area with PB = 5.6 cm. Find the height of ABCD and the area of PBE.
32After constructing parallelogram ABCD with AB = 7 cm, BC = 5 cm and \( \angle ABC = 120^\circ \), and the rectangle ABFE equal to it in area, find BF and the area of ABCD.
33In a circle with centre O, central angle \( \angle BOC \) and inscribed angle \( \angle BDC \) stand on arc BC. Write their relation, verify it experimentally, and find x if \( \angle BOC = 7x^\circ \) and \( \angle BDC = (3x + 5)^\circ \).
34ABCD is a cyclic quadrilateral in a circle with centre O. Write the relation of \( \angle ABC \) and \( \angle ADC \), prove \( \angle ADC = \tfrac{1}{2}\angle AOC \), and find \( \angle AOC \) if \( \angle ABC = 120^\circ \).
35P, Q, R and S are on a circle; PSA and QRA are straight lines and AP = AQ. Name the inscribed angles on arc PQ; if \( \angle PSQ = 60^\circ \), find \( \angle PRQ \); and prove PR = QS.
36O is the centre and SH a diameter; U and I are on the circle, S, U and L are on one line, and LO is perpendicular to SH. Find \( \angle SUH \), prove \( \angle SIU = \angle OLS \), and find \( \angle SIU \) if \( \angle USH = 50^\circ \).
37O is the centre and PR and QS are diameters; \( \angle PQS = y \), \( \angle QSR = 63^\circ \) and \( \angle PRQ = x \). Find \( \angle PQR \) and \( \angle POS \), and prove \( x + y = 90^\circ \) and that triangle QOR is isosceles.
38In a cyclic quadrilateral ABCD, \( \angle A : \angle C = 2 : 3 \). Find \( \angle A \) and \( \angle C \).
39Can a parallelogram that is not a rectangle be cyclic? Why?
40An exterior angle of a cyclic quadrilateral is 105°. Find the interior opposite angle and the interior angle at the same vertex.