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If two opposite sides of a cyclic quadri...

If two opposite sides of a cyclic quadrilateral are equal, then the other two sides are parallel.

Text Solution

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Given: In cyclic quadrilateral `ABCD`, `AD=BC`.
To Prove: `AB∥BC`
Construction: Join `BD`.
Proof: `AD=BC` [Given]
`Arc DA=Arc BC`
`∠1=∠2` [Equal chord subtend equal angle at the circumference of the circle]
But `∠1` and `∠2` are alternate interior angles.
`∴AB` is parallel to `CD`.
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Knowledge Check

  • If the opposite sides of a cyclic quadrilateral are equal and the sum of two of the adjacent sides is 10 cm, find the maximum possible area of the quadrilateral (in sq. cm).

    A
    A)100
    B
    B)25
    C
    C)24
    D
    D)data insufficient

  • If the opposite sides of a quadrilateral and also its diagonals are equal, then each of the angles of the quadrilateral is

    A
    `90^(@)`
    B
    `120^(@)`
    C
    `100^(@)`
    D
    `60^(@)`

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    Theorem 8.3 : If each pair of opposite sides of a quadrilateral is equal, then it is a parallelogram.

    If two sides of a cyclic quadrilateral are parallel,prove that the remaining two sides are equal and the diagonals are also equal.OR A cyclic trapezium is isosceles and its diagonals are equal.

    Prove that the bisectors of the angles formed by producing the opposite sides of a cyclic quadrilateral (provided that they , are not parallel), intersect at right angles.

    The midpoint of two opposite sides of a quadrilateral and the midpoint of the diagonals are the vertices of a parallelogram. Prove that using vectors.

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