if the sum of a pair of opposite angles of a quadrilateral is 180o the quadrilateral is cyclic

given: ABCD is a quadrilateral such that m∠B+m∠D=180

TPT: ABCD is a cyclic quadrilateral.

proof: this we can show by the contradiction,

if possible let ABCD is not cyclic quadrilateral. draw a circle thriugh points A, B and C .

let circle meets AD or extended AD at D'.

therefore ABCD' is cyclic.

therefore ∠ABC+∠AD'C=180......(1) [sum of opposite angles of a cyclic quadrilateral are 180 deg]

but ∠ABC+∠ADC = 180 deg .........(2) [given]

thus from (1) and (2) ∠AD'C = ∠ADC.....(3)

but exterior angle can not be equal to interior angle. as it is equal to sum of two opposite angles.

∠AD'C=∠ADC+∠DAD'

therefore ∠DAD' = 0 thus D' coincides with D.

thus ABCD is a cyclic quadrilateral.

hope this helps you.