Prove that if a pair of opposite angles of a quadrilateral is supplementary, then the quadrilateral is cyclic. [4 MARKS]
Concept: 2 Marks
Application: 2 Marks
Consider a quadrilateral ABPQ, such that ∠ABP+∠AQP=180∘ and ∠QAB+∠QPB=180∘
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.
∠QAB+∠QRB=180∘ [opposite angles of cyclic quadrilateral.]
But this cannot be true since ∠QRB=∠QPB+∠RBP (exterior angle of the triangle)
∴ 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.
∴ ABPQ is a cyclic quadrilateral.