Question

Prove that ‘Opposite angles of a cyclic quadrilateral are supplementary’.

Answer 1

Given: `square`ABCD is a cyclic quadrilateral.

To prove: ∠BAD + ∠BCD = 180º 

∠ABC + ∠ADC = 180º

Proof: Arc BCD is intercepted by the inscribed ∠BAD.

∠BAD = `1/2` m(arc BCD)     ...(i) [Inscribed angle theorem]

Arc BAD is intercepted by the inscribed ∠BCD.

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

From (1) and (2) we get

∠BAD + ∠BCD = `1/2` [m(arc BCD) + m(arc DAB)]

∴ (∠BAD + ∠BCD) = `1/2 xx 360^circ`    ...[Completed circle]

= 180°

Again, as the sum of the measures of angles of a quadrilateral is 360°

∴ ∠ADC + ∠ABC = 360° – [∠BAD + ∠BCD]

= 360° – 180°

= 180°

Hence, the opposite angles of a cyclic quadrilateral are supplementary.