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Question
Prove that "Opposite angles of a cyclic quadrilateral are supplementary".
Solution
Given:- □ABCD is a cyclic quadrilateral.
To prove:- ∠BAD + ∠BCD = 180º and ∠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.
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