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Circles

Short Answer Type

71. ABC is an isosceles triangle with AB = AC. A circle through B and C intersects AD and AC at D and E respectively. Prove that BC || DE.

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72. Prove that the circle drawn on any equal side of an isosceles triangle as diameter, bisects the third side.

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73.

In the figure, D is the centre of a circle. Prove that
2(∠XZY + ∠YXZ) = ∠XPZ]

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Long Answer Type

74. Prove that the opposite angles of an isosceles trapezium are supplementary.

Given: ABCD is trapezium in which AD = BC
To prove: Opposite angles of ABCD are supplementary.
Construction: Draw BE || AD
Proof: ∵ Quadrilateral ABED is a parallelogram

AB || DC (Given) AD || BE (By construction).
| A quadrilateral is a parallelogram if its any pair of opposite sides are parallel and equal.
∴ ∠BAD = ∠BED
| Opposite angles of a parallelogram are equal
and AD = BE
| Opposite sides of a parallelogram are equal
But AD = BC | Given
∴ BE = BC
∴ ∠BEC = ∠BCE
| Angles opposite to equal sides of a triangle are equal
∴ ∠BEC + ∠BED = 180° | Linear pair
⇒ ∠BCE + ∠BED = 180°
⇒ ∠BAD + ∠BED = 180°
⇒ ∠ADE + ∠ABE = 180°
⇒ Opposite angles of ABCD are supplementary.

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75. In the figure, O is the centre of the circle. BD = OD and CD ⊥ AB. Find ∠CAB.

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76. If the bisectors of the opposite angles of a cyclic quadrilateral ABCD intersect the circle circumscribing it at the points P and Q, prove that PQ is a diameter of the circle.

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