Question

Prove that the “the opposite angles of the cyclic quadrilateral are supplementary”.

Answer 1

arc ABC is intercepted by the inscribed angle ∠ADC.

∴ ∠ADC = `1/2` m(arc ABC)    ......(i) [Inscribed angle theorem]

Similarly, ∠ABC is an inscribed angle.

It intercepts arc ADC.

∴ ∠ABC =`1/2` m(arc ADC)    ......(ii) [Inscribed angle theorem]

∴ ∠ADC + ∠ABC = `1/2` m(arc ABC) + `1/2` m(arc ADC)  ....[Adding (i) and (ii)]

∴ ∠D + ∠B = `1/2` m (arc ABC) + m(arc ADC)]

∴ ∠B + ∠D = `1/2 xx 360^circ`    ......[arc ABC and arc ADC constitute a complete circle]

= 180°

∴ ∠B + ∠D = 180°

Similarly we can prove,

∠A + ∠C = 180°