Question

For proving a quadrilateral to be cyclic. , We have to prove one pair of opposite angles or both pairs of opposite angles to be 180

Answer 1

Dear Student,

If you prove even a single pair of opposite angles in a quadrilateral to be supplementary, the quadrilateral is a cyclic quadrilateral.
Because if one pair is supplementary, that is the sum of opposite angles is 180 degrees, the sum of another pair is also 180 degrees.

$$\begin{gathered} (Sumofallanglesofaquadrilateral=360° \\ Let\angle A,\angle B,\angle Cand\angle Dbethe4angles. \\ Let\angle A+\angle C=180° \\ \angle A+\angle B+\angle C+\angle D=360° \\ \left(\angle A+\angle C\right)+\left(\angle B+\angle D\right)=360° \\ 180°+\left(\angle B+\angle D\right)=360° \\ \left(\angle B+\angle D\right)=360°-180° \\ \angle B+\angle D=180° \\ So,ifonepairofoppositeanglesissupplementary,theanotheroneisalso) \end{gathered}$$

Regards