Question

prove sum of opposite angle of cyclic quadrilateral is 180

Answer 1

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Here is ur answer!!

⇒ Given : Let ABCD is cyclic quadrilateral.

⇒ To prove : ∠A + ∠C = 180°  and ∠B + ∠D = 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)X360° = 180°

√ Similarly ∠B + ∠D = 180°

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★★★hope this may help u!!★★★

Answer 2

Given :

 ABCD is cyclic quadrilateral.

To prove :

 ∠A + ∠C = 180°  and ∠B + ∠D = 180°.

Construction :

Join OB and OD.

Proof :

 ∠BOD = 2 ∠BAD

(By Theorem angle subtend at centre and point in circle with same line

segment)

∠BAD = 1/2∠ BOD            .......1

Also,

∠BCD = 1/2 ∠DOB           ........2

Adding 1 and 2

∠BAD + ∠BCD =1/2(∠ BOD + ∠DOB)
                 
∠BAD + ∠BCD = (1/2) × 360° = 180°

(Because in Fig ∠BOD and ∠DOB are to be assumed opposite angles)

So, 

∠A + ∠C = 180°

And so,

 ∠B + ∠D = 180°

And from this The theorem gets proved.


Figure :

See Below ↓↓↓

☺ Hope this Helps ☺