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Cyclic Quadrilaterals

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A quadrilateral ABCD is called cyclic if all the four vertices of it lie on a circle in fig.  A quadrilateral is said to be a cyclic quadrilateral, if there is a circle passing through all its four vertices.
`=>`∠A + ∠C = 180° 
also ∠B + ∠D = 180°.

theorem

Theorem: The sum of either pair of opposite angles of a cyclic quadrilateral is 180º. 
Given : A cyclic quadrilateral ABCD . 
To Prove : ∠BAD + ∠BCD = ∠ABC + ∠ADC = 180º. 
Construction : Draw AC and DB.
Proof : ∠ACB = ∠ ADB  and 
∠BAC = ∠BDC [Angles in the same segment]
`therefore` ∠ACB + ∠BAC = ∠ADB +∠BDC = ∠ADC
Adding ∠ABC on both the sides, we get
∠ACB + ∠BAC +∠ABC = ∠ADC +∠ABC 
But ∠ACB +∠BAC +∠ABC = 180º.   [sum of the angles of a triangle]
`therefore` ∠ ADC + ∠ABC =180º. 
`therefore` ∠BAD +∠BCD = 360º  - (∠ADC + ∠ABC) = 180º. 
Hence proved.

Theorem: If the sum of a pair of opposite angles of a quadrilateral is 180º, the quadrilateral is cyclic.

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