# 21. Prove that If one side of the cyclic quadrilateral is produced then the exterior angle is equal to the interior opposite angle.

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2 21. Prove that If one side of the cyclic quadrilateral is produced then the exterior angle is equal to the interior opposite angle. 22. Prove that If two sides of a cyclic quadrilateral are parallel, then the remaining two sides are equal and the diagonals are also equal. 23. Prove that If two opposite sides of cyclic quadrilateral are equal, then the other two sides are parallel. 24. Prove that If two non parallel sides of a trapezium are equal, it is cyclic. 25. Prove that The sum of the angles in the four segments exterior to a cyclic quadrilateral is equal to 6 right angles. 26. Two circles with centres A and B intersect at C and D. Prove that ACB = ADB. 27. Bisector AD of AC of ABC passes through the centre of the circumcircle of ABC. Prove that AB = AC. 28. In the below figure A, B and C are three points on a circle such that angles subtended by the chords AB and AC at the centre O are 80 0 and respectively. Determine BAC. 29. In the above right-sided figure, P is the centre of the circle. Prove that XPZ = 2 ( XZY + YXZ). 30. Prove that the midpoint of the hypotenuse of a right triangle is equidistant from its vertices. 31. In the below figure ABCD is a cyclic quadrilateral, O is the centre of the circle. If BOD = 160 0, find BPD. 32. Prove that in a triangle if the bisector of any angle and the perpendicular bisector of its opposite side intersect, they will intersect on the circumcircle of the triangle. Page 2

3 33. The diagonals of a cyclic quadrilateral are at right angles. Prove that perpendiculars from the point of their intersection on any side when produced backward bisect the opposite side. 34. If two circles intersect at two points, prove that their centres lie on the perpendicular bisector of the common chord. 35. If two intersecting chords of a circle make equal angles with the diameter passing through their point of intersection, prove that the chords are equal. 36. Two circles of radii 5 cm and 3 cm intersect at two points and the distance between their centres is 4 cm. Find the length of the common chord. 37. If two equal chords of a circle intersect within the circle, prove that the segments of one chord are equal to corresponding segments of the other chord. 38. If two equal chords of a circle intersect within the circle, prove that the line joining the point of intersection to the centre makes equal angles with the chords. 39. In the below figure, AB is a diameter of the circle, CD is a chord equal to the radius of the circle. AC and BD when extended intersect at a point E. Prove that AEB = In the above right-sided figure, ABCD is a cyclic quadrilateral in which AC and BD are its diagonals. If DBC = 55 and BAC = 45, find BCD. 41. Prove that the quadrilateral formed (if possible) by the internal angle bisectors of any quadrilateral is cyclic. 42. ABCD is a cyclic quadrilateral whose diagonals intersect at a point E. If DBC = 70, BAC is 30, find BCD. Further, if AB = BC, find ECD. 43. If diagonals of a cyclic quadrilateral are diameters of the circle through the vertices of the quadrilateral, prove that it is a rectangle. 44. Two circles intersect at two points A and B. AD and AC are diameters to the two circles. Prove that B lies on the line segment DC. 45. Prove that the quadrilateral formed (if possible) by the internal angle bisectors of any quadrilateral is cyclic. 46. If the non-parallel sides of a trapezium are equal, prove that it is cyclic. Page 3

4 47. Two circles intersect at two points B and C. Through B, two line segments ABD and PBQ are drawn to intersect the circles at A, D and P, Q respectively. Prove that ACP = QCD. 48. If circles are drawn taking two sides of a triangle as diameters, prove that the point of intersection of these circles lie on the third side. 49. Prove that the circle drawn with any side of a rhombus as diameter, passes through the point of intersection of its diagonals. 50. In the adjoining figure, A, B, C and D are four points on a circle. AC and BD intersect at a point E such that BEC = 130 and ECD = 20. Find BAC. 51. In the above right-sided figure, PQR = 100, where P, Q and R are points on a circle with centre O. Find OPR. 52. ABCD is a parallelogram. The circle through A, B and C intersect CD (produced if necessary) at E. Prove that AE = AD. 53. AC and BD are chords of a circle which bisect each other. Prove that (i) AC and BD are diameters, (ii) ABCD is a rectangle. 54. A chord of a circle is equal to the radius of the circle. Find the angle subtended by the chord at a point on the minor arc and also at a point on the major arc. 55. Prove that the circle drawn with any side of a rhombus as a diameter, passes through the point of its diagonals. 56. Bisectors of angles A, B and C of a triangles ABC intersect its circumcircle at D, E and F respectively. Prove that the angles of DDEF are A, B and C 57. Prove that the line of centres of two intersecting circles subtends equal angles at the two points of intersection. Page 4

5 58. In the adjoining Fig., ABC = 69, ACB = 31, find BDC. 59. In the above right-sided figure, A,B and C are three points on a circle with centre O such that BOC = 30 and AOB = 60. If D is a point on the circle other than the arc ABC, find ADC. 60. In the below figure, AB and CD are two equal chords of a circle with centre O OP and OQ are perpendiculars on chords AB and CD, respectively. If POQ =, 150 then find APQ. 61. In the above right sided figure, if OA = 5 cm, AB = 8 cm and OD is perpendicular to AB, then find CD. 62. Two chords AB and CD of lengths 5 cm and 11 cm respectively of a circle are parallel to each other and are on opposite sides of its centre. If the distance between AB and CD is 6 cm, find the radius of the circle. 63. Two congruent circles intersect each other at points A and B. Through A any line segment PAQ is drawn so that P, Q lie on the two circles. Prove that BP = BQ. 64. In any triangle ABC, if the angle bisector of A and perpendicular bisector of BC intersect, prove that they intersect on the circumcircle of the triangle ABC. 65. If arcs AXB and CYD of a circle are congruent, find the ratio of AB and CD. 66. If the perpendicular bisector of a chord AB of a circle PXAQBY intersects the circle at P and Q, prove that arc PXA Arc PYB. 67. A, B and C are three points on a circle. Prove that the perpendicular bisectors of AB, BC and CA are concurrent. Page 5

6 68. AB and AC are two equal chords of a circle. Prove that the bisector of the angle BAC passes through the centre of the circle. 69. In the below figure, if OAB = 40 0, then find ACB 70. In the above right sided figure, if DAB = 60 0, ABD = 50 0 then find ACB. 71. In the below figure, BC is a diameter of the circle and BAO = 60 then find ADC In above right sid d figure, AOB = 90 0 and ABC = 30 0, then find CAO 73. The lengths of two parallel chords of a circle are 6 cm and 8 cm. If the smaller chord is at distance 4 cm from the centre, what is the distance of the other chord from the centre? 74. A, B, C D are four consecutive points on a circle such that AB = CD. Prove that AC = BD. 75. If a line segment joining mid-points of two chords of a circle passes through the centre of the circle, prove that the two chords are parallel. 76. ABCD is such a quadrilateral that A is the centre of the circle passing through B, C and D. Prove that CBD + CDB = 1 2 BAD 77. O is the circumcentre of the triangle ABC and D is the mid-point of the base BC. Prove that BOD = A. 78. On a common hypotenuse AB, two right triangles ACB and ADB are situated on opposite sides. Prove that BAC = BDC. Page 6

7 79. In the below figure, AOC is a diameter of the circle and arc(axb) = 1 arc(byc). Find BOC In the above right sided figure, ABC = 45 0, prove that OA OC. 81. Two chords AB and AC of a circle subtends angles equal to 90 and, 150 respectively at the centre. Find BAC, if AB and AC lie on the opposite sides of the centre. 82. If BM and CN are the perpendiculars drawn on the sides AC and AB of the triangle ABC, prove that the points B, C, M and N are concyclic. 83. If a line is drawn parallel to the base of an isosceles triangle to intersect its equal sides, prove that the quadrilateral so formed is cyclic. 84. If a pair of opposite sides of a cyclic quadrilateral are equal, prove that its diagonals are also equal. 85. The circumcentre of the triangle ABC is O. Prove that OBC + BAC = A chord of a circle is equal to its radius. Find the angle subtended by this chord at a point in major segment. 87. In the below figure, ADC = 130 and chord BC = chord BE. Find CBE. 88. In the above right sided figure, ACB = Find OAB. 89. A quadrilateral ABCD is inscribed in a circle such that AB is a diameter and ADC = Find BAC. 90. Two circles with centres O and O intersect at two points A and B. A line PQ is drawn parallel to OO through A(or B) intersecting the circles at P and Q. Prove that PQ = 2 OO Page 7

8 91. In the below figure, AOB is a diameter of the circle and C, D, E are any three points on the semicircle. Find the value of ACD + BED. 92. In the above right sided figure, AB = 30 0 and OCB = Find BOC and AOC. 93. In the below figure, O is the centre of the circle, BCO = 30 0, find x and y. 94. In the above right sided figure, O is the centre of the circle BD = OD and CD AB. Find CAB. 95. Let the vertex of an angle ABC be located outside a circle and let the sides of the angle intersect equal chords AD and CE with the circle. Prove that ABC is equal to half the difference of the angles subtended by the chords AC and DE at the centre. Page 8

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