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Prove that ‘The Opposite Angles of a Cyclic Quadrilateral Are Supplementary’. - Geometry


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

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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.

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