Question

Prove that the sum of either pair of the opposite angles of a cyclic quadrilateral is 180° or
the opposite angles of a cyclic quadrilateral are supplementary.​

Answer 1

Step-by-step explanation:

Given : ABCD is a cyclic quadrilateral of a circle with centre at O.

To prove : ∠BAD+∠BCD=180°

∠ABC+∠ADC=180°

Chord AB

∠5=∠8.....(1) [Angle in same segment are equal]

Chord BC

∠1=∠6.....(2) [Angle in same segment are equal]

Chord CD

∠2=∠4.....(3) [Angle in same segment are equal

Chord AD

∠7=∠3.....(4) [Angle in same segment are equal]

By angle sum property of qudrilateral

∠A+∠B+∠C+∠D=360°

∠1+∠2+∠3+∠4+∠7+∠8+∠5+∠6=360°

(∠1+∠2+∠7+∠8)+(∠3+∠4+∠5+∠6)=360°

(∠1+∠2+∠7+∠8)+(∠7+∠2+∠8+∠1)=360°

[From (1), (2) ,(3) and (4)]

2(∠1+∠2+∠7+∠8)=360°

∠1+∠2+∠7+∠8=180°

(∠1+∠2)+(∠7+∠8)=180°

∠BAD+∠BCD=180°

Similarly,

∠ABC+∠ADC=180°

Hence proved.

Answer 2

Answer:

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Step-by-step explanation:

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