Which of the following cannot be a cyclic quadrilateral?

Square

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Rectangle

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Parallelogram

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Trapezium

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Solution

The correct options are
C
Parallelogram

D
Trapezium

A quadrilateral is cyclic only when the sum of either pair of opposite angles is $180^{\circ}$ .
Each angle of a square and rectangle is $90^{\circ}$ . Hence both pairs of opposite angles are supplementary.
In parallelograms, the sum of adjacent angles is $180^{\circ}$ . Only in the case of rectangles and squares, the opposite angles can also be $180^{\circ}$ . For other parallelograms, this is impossible since the sum of one pair of opposite angles will be an obtuse angle and the sum of the other pair of opposite angles will be an acute angle.
A trapezium can also be a cyclic quadrilateral provided it is an isosceles trapezium in which the pair of non-parallel sides are equal.

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ii) It can be a square

iii) It can be a rectangle

(iv) It can be a trapezium

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