Question

Prove that if a pair of opposite angles of a quadrilateral is supplementary, then the quadrilateral is cyclic. [4 MARKS]

Answer 1

Concept: 2 Marks
Application: 2 Marks

Consider a quadrilateral ABPQ, such that $\angle ABP+\angle AQP={180}^{\circ }$ and $\angle QAB+\angle QPB={180}^{\circ }$

To prove: The points A, B, P and Q lie on the circumference of a circle.

Assume that point P does not lie on a circle drawn through points A, B and Q. Let the circle cut QP at point R. Join BR.

$\angle QAB+\angle QRB={180}^{\circ }$ [opposite angles of cyclic quadrilateral.]

$\angle QAB+\angle QPB={180}^{\circ }$ [given]

$\therefore \angle QRB=\angle QPB$

But this cannot be true since $\angle QRB=\angle QPB+\angle RBP$ (exterior angle of the triangle)

$\therefore$ Our assumption that the circle does not pass through P is incorrect and A, B, P and Q lie on the circumference of a circle.

$\therefore$ ABPQ is a cyclic quadrilateral.