Question

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

Answer 1

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Given:- Let ABCD is a cyclic quadrilateral.

To prove:- ∠A + ∠C = 180° and ∠B + ∠C = 180°

Construction :- Join OB and OD.

Proof :- ∠BOD = 2∠BAD

            ∠BAD = 1/2∠BOD

Similarly ∠BCD = 1/2∠DOB

∠BAD + ∠BCD = 1/2∠BOD + 1/2∠DOB

                        = 1/2(∠BOD + ∠DOB)

                        = 1/2(360°)

                        = 180°

Similarly ∠B +∠D = 180° (Hence Proved)

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Answer 2

Theorem - The opposite angles of a cyclic quadrilateral are supplementary.

Given - ABCD is a cyclic quadrilateral

To prove - Opposite angles to be supplementary.

Construction - Join OB and OD

Proof -

We know,

Angle (BOD) = 2*Angle(BAD)...(1)

(By property of circle)

Similarly,

Angle(DOB) = 2*Angle(BCD).... (2)

Adding (1) and (2),

Angle (BAD) + Angle (BCD) = 1/2*Angle(BOD) + 1/2*Angle(DOB)

= 1/2*{Angle(BOD) + Angle(DOB)}

= 1/2*360

= 180°

Similarly,

Angle(B) + Angle(D) = 180°

Hence proved.