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If non-parallel sides of a trapezium are...

If non-parallel sides of a trapezium are equal, prove that it is cycli.

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Given : Trapezium ABCD in which `AB||CD` and ` AD||BC`.
To Prove : `angle 1+angle2 =180^@`
Construction: Draw `CE||DA`.
Proof : Since, ABCD is a trapezium in which
`AB||DC`
`rArrAE||DC`
and `CE||DA`
`thereforesquare AECD` is a parallelogram (`because` pairs of opposite sides of a parallelogram )
`thereforeAD=CE` (opposite sides of a parallelogram are equal)
But `AD=CB `(given)
`therefore =CB`
`rArrangle2=angle3` (angles opposite to equal sides are eqaul ).....(1)
Now, `angle3+angle4=180^@` (L.P.A.)
`rArrangle2+angle4=180^@` [form (1)]
`rArrangle2+angle1 =180^@`
Hence, ABCD is cyclic.
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