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.

Explanation

Given: Two sides of a cyclic quadrilateral are parallel.
To prove: The remaining two sides are equal and the diagonals are also equal.
Proof: Since `ABCD` is cyclic quadrilateral, opposite angles are supplementary.
`=>angleB+angleD=180^@`......(i)
`AB` parallel to `CD` and `BC` is a transversal,
`=>angleABC+angleBCD=180^@` (Sum of interior angles`=180^@`)
`=>angleB+angleC=180^@`............ (ii)
From equation (i) and equation (ii), we get:
`angleB+angleD=angleB+angleC`
`=>angleD=angleC`......(iii)
In triangle`ADC` and `triangleBCD`,
`angleADC=angleBCD` [From equation (iii)]
`CD=CD` [Common]
`angleDAC=angleCBD`
`=>triangleADC~=triangleBCD`
`AD=BC` and `AC=BD`
`therefore` The remaining two sides are equal and the diagonals are also equal.