The opposite angles of a cyclic quadrilateral are supplementary.

The opposite angles of a cyclic quadrilateral are supplementary.

Given: □ ABCD is a cyclic quadrilateral.

To Prove: ∠ A + ∠ C = 180⁰ and ∠ B + ∠ D = 180⁰

Solution:

∠ A = ½ m(arc BCD)  --------------eq. no. (1) [Inscribed angle theorem]

∠ C = ½ m (arc BAD)   -------------- eq. no. (2) [Inscribed angle theorem]

Adding equations (1) and (2)

∴ ∠ A + ∠ C = ½ m(arc BCD) + ½ m(arc BAD)

∴ ∠ A + ∠ C = ½ [m(arc BCD) + m(arc BAD)]

∴ ∠ A + ∠ C = ½ [Measure of a circle]

∴ ∠ A + ∠ C = ½ [3600]

∴ ∠ A + ∠ C = 1800

Similarly, we can prove that,

∠ B + ∠ D = 1800