Question

Abcd is such a quadrilateral that a is the center of a circle passing through b,c,d.prove that angle CBD +angle CDB=1/2angle BAD

Answer 1

We will refer to the circle passing through B, C, and D as circle A. 

∠CBD is an angle inscribed in circle A, intercepting arc CD. 
So m∠CBD is half the measure of arc CD. 

∠CDB is an angle inscribed in circle A, intercepting arc BC. 
So m∠CDB is half the measure of arc BC. 

So the sum of the measures of ∠CBD and ∠CDB is 
half the measure of arc BD (passing through C). 

But the central angle for that arc, on which its measure is based, is ∠BAD. 
So 
m∠CBD + m∠CDB = (1/2) m∠BAD
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Answer 2

here is your answer

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