61.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 (see figure). Prove that ∠ACP = ∠QCD.
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62.If circles are drawn taking two sides of a triangle as diameters, prove that the point of intersection of these circles lies on the third side.
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63.
ABC and ADC are two right triangles with common hypotenuse AC. Prove that ∠ CAD = ∠ CBD.
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64.
Prove that a cyclic parallelogram is a rectangle.
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Long Answer Type
65.ABCD is a cyclic trapezium with AD || BC. If ∠B = 70°, determine other three angles of the trapezium.
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66.In figure, a diameter AB of a circle bisects a chord PQ. If AQ || PB, prove that the chord PQ is also a diameter of the circle.
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Short Answer Type
67.In figure, AB = CD. Prove that BE = DE and AE = CE, where E is the point of intersection of AD and BC.
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Long Answer Type
68.ABCD is a cyclic quadrilateral with AD || BC. Prove that AB = DC.
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Short Answer Type
69.In the given figure, ABCD is a cyclic quadrilateral whose diagonals intersect at P. If ∠DBC = 70° and ∠BAC = 30°, find ∠BCD.
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70.In the given figure, ABCD is a cyclic quadrilateral whose side AB is a diameter of the circle through A, B, C, D. If ∠ADC = 140°, find ∠BAC.
Short Answer Type
Long Answer Type
