Prove that ‘the opposite angles of a cyclic quadrilateral are supplementary’.
Solution
Given:- □ABCD is cyclic quadrilateral
To prove:- ∠BAD + ∠BCD = 180º
and ∠ABC + ∠ADC = 180º
Proof :-
Arc BCD is intercepted by the inscribed ∠BAD.
`therefore angle"BAD"=1/2"m"("arc BCD")..........(1)`
(Inscribed angle theorem)
Arc BAD is intercepted by the inscribed ∠BCD.
`therefore angle"BCD"=1/2"m" ("arc DAB")..........(2)`
(Inscribed angle theorem)
From (1) and (2) we get
∠BAD + ∠BCD = 1/2[M(arc BCD) + M(arc DAB)]
= (1/2)*360°
= 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.