If the sum of a pair of opposite angels of a quadrilateral is 180; the quadrilateral is cyclic

If the sum of a pair of opposite angels of a quadrilateral is 180; the

1.	If the sum of a pair of opposite angels of a quadrilateral is 180; the quadrilateral is cyclic
Answer» Given the sum of pair of opposite angles of a quadrilateral is 180° we have to prove that the quadrilateral is cyclic.Let us assume that the quadrilateral ABCD is not cyclic i.e Let the point D does not lie on the circle which makes the quadrilateral non-cyclic. Now, let us do a construction such that join CD\' where D\' is the point of intersection of side AD with the circle.Now, ABCD\' is cyclic⇒ ∠3 + ∠4 = 180°Now, it is given that the sum of pair opposite angles of a quadrilateral ABCD is 180°Therefore, ∠2 + ∠4 = 180°From above two equations we get∠3 + ∠4 = ∠2 + ∠4⇒ ∠3 = ∠2Now, in triangle CDD\', by external angle property∠3 = ∠1 + ∠2⇒ ∠1 = 0 , hence the side CD\' and CD coincides⇒ Point D lies on circleHence, our supposition is wrong quadrilateral ABCD is cyclic.