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Question

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


Solution

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.]

QAB+QPB=180 [given]

QRB=QPB

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.


Mathematics

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