Prove that the sum of either pair of the opposite angles of a cyclic quadrilateral is 180°
the opposite angles of a cyclic quadrilateral are supplementary.
iel D.
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Answer:
Given : ABCD is a cyclic quadrilateral of a circle with centre at O.
To prove : ∠BAD+∠BCD=180
o
∠ABC+∠ADC=180
o
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
o
∠1+∠2+∠3+∠4+∠7+∠8+∠5+∠6=360
o
(∠1+∠2+∠7+∠8)+(∠3+∠4+∠5+∠6)=360
o
(∠1+∠2+∠7+∠8)+(∠7+∠2+∠8+∠1)=360
o
[From (1), (2) ,(3) and (4)]
2(∠1+∠2+∠7+∠8)=360
o
∠1+∠2+∠7+∠8=180
o
(∠1+∠2)+(∠7+∠8)=180
o
∠BAD+∠BCD=180
o
Similarly,
∠ABC+∠ADC=180
o
Hence proved.
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