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ABCD is such a quadrilateral that A is t...

`ABCD` is such a quadrilateral that A is the centre of the circle passing through `B, C and D`. Prove that `/_CBD+ /_CDB =1/2 /_BAD`.

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With A as centre of AB as radius , draw a circle passing through the points B,C and D.
Take a point E onteh circle outsidearc BCD. Join BE, Deand BD
Clearly, `/_ BAD = 2/_ BED` …..(i)
Now, EBCD is a cyclic quadrilateral.
`:. /_ BED + /_ BCE = 180^(@)`
`implies /_ BCD = 180^(@) - /_ BED`
`implies /_ BCD = 180^(@) - (1)/(2) /_ BAD` ....(ii) [using (i) ]
In `Delta BCD` , we have
`/_ CBD + /_ BCD + /_ CDB = 180^(@)`
`implies /_ CBD +/_ CDB = 180^(@) - /_ BCD`
`= 180^(@) - ( 180^(@) - (1)/(2) /_ BAD )= (1)/(2) /_ BAD ` [ using (ii) ].
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