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Theorem: Opposite angles of a cyclic quadrilateral are supplementry. Fill in the blanks and complete the following proof. Given: □ ABCD is cyclic. To prove: ∠B + ∠D = 180° ∠A + ∠C = 180°

Answer»

Proof: 

arc ABC is intercepted by the inscribed angle ∠ADC. 

∴ ∠ADC m(arcABC) (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(arcABC) + 1/2 m(arc ADC) [Adding (i) and (ii)] 

∴ ∠D + ∠B = 1/2 m(areABC) + m(arc ADC)] 

∴ ∠B + ∠D = 1/2 × 360° [arc ABC and arc ADC constitute a complete circle] = 180°

∴ ∠B + ∠D = 180° 

Similarly we can prove, 

∠A + ∠C = 180°


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