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Circles

Short Answer Type

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.

Given: ABCD is a cyclic quadrilateral with AD || BC.
To Prove: AB = DC.

Construction: Join AC.
Proof: ∵ AD || BC
and AC intersects them
∴ ∠ACB = ∠CAD | Alt. Int. ∠s
∴ arc AB ≅ arc CD
| Arcs opposite to equal angles are congruent
∴ chord AB = chord CD
| If two arcs of a circle are congruent, then their corresponding chords are equal
⇒ AB = CD
⇒ AB = DC.
Aliter:
Given: ABCD is a cyclic quadrilateral with AD || BC.
To Prove: AB = DC.

Construction: Draw DE || AB.
Proof: AD || BC    | Given
⇒ AD || BE
AB || DE    | By const.
∴ Quadrilateral ADEB is a parallelogram
∴ AB = DE    ...(1)
| Opp. sides of a || gm are equal
and    ∠BAD = ∠BED    ...(2)
| Opp. ∠s of a || gm are equal
∵ ABCD is a cyclic quadrilateral
∴ ∠BAD + ∠BCD = 180°    ...(3)
| ∵ Opposite angles of a cyclic quadrilateral are supplementary
∠BED + ∠CED = 180° | Linear pair
⇒ ∠BAD + ∠CED = 180°    ...(4)
| From (2)
From (3) and (4),
∠BCD = ∠CED
⇒    ∠ECD = ∠CED
∴ DE = DC
| Sides opp. to equal angles
⇒    AB = DC.    | From(1)

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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.

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