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In the given figure, prove that x=alpha+...

In the given figure, prove that `x=alpha+beta+gamma`.

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Join B and D and produce BD to E.
Let `angleABD=p^(@),angleCBD=q^(@)and" let "angleADE=s^(@)andangleCDE=t^(@)`.
Then, `p+q=betaands+t=x`.
Now, side BD of `DeltaABD` has been produced to E.
`:.s=p+alpha" "....(i)`
Again, side BD of `DeltaCBD` has been produced to E.
`:.t=q+gamma`.
Adding the corresponding sides of (i) and (ii), we get
`s+t=(p+q)+alpha+gamma`
`impliesx=beta+alpha+gamma." "[:'s+t=xandp+q=beta]`
Hence, `x=alpha+beta+gamma`.
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