Question

Can u give me the proof of the theorem: If the sum of a pair of opposite angles of a quadrilateral is 180, the quadrilateral is cyclic.

Answer 1

Theorem : Sum of the opposite angles of a cyclic quadrilateral is 180°.
Given : A cyclic quadrilateral ABCD.
To prove : ∠BAD + ∠BCD = ∠ABC + ∠ADC = 180°
Construction : Draw AC and DB
Proof : ∠ACB = ∠ADB
and ∠BAC = ∠BDC                                                   [Angles in the same segment]
∴ ∠ACB + ∠BAC = ∠ADB + ∠BDC = ∠ADC
Adding ∠ABC on both the sides, we get
∠ACB + ∠BAC + ∠ABC = ∠ADC + ∠ABC
But ∠ACB + ∠BAC + ∠ABC = 180°                           [Sum of the angles of a triangle]
∴ ∠ADC + ∠ABC = 180°
∴ ∠BAD + ∠BCD = 360° - (∠ADC + ∠ABC) = 180°.
Hence proved.
Converse of this theorem is also true.
If a pair of opposite angles of a quadrilateral is supplementary, then the quadrilateral is cyclic.