Question

Prove the following theorem:

Opposite angles of a 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°