# Solve this: Q.19. ABCD is a parallelogram, bisectors of angles A and B meet at E which lies on DC. Prove that AB = 2 AD.

nhgnghnytu
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ABCD is a parallelogram , SO we know

AB  =  CD  and BC  =  DA
And
AB  | | CD   and BC | | DA Here ∠ DAE   =  ∠ EAB ( As given AE is angle bisector of ∠ BAD )
And
∠ CBE  =  ∠ EBA    (  As given BE is angle bisector of ∠ ABC )

Now we take AE as transversal line and we know AB | | CD
So
∠ EAB  =  ∠ DEA   (  As they are alternate interior angles )
So,
∠ DEA   =  ∠ DAE   ( As we know ∠ DAE  = ∠ EAB )
So,
AD  = DE( As we know base angle theorem , if base angles of any triangle is equal than opposite sides are also equal )

And
Now we take AE as transversal line and we know AB | | CD
So
∠ EBA  =  ∠ CEB (  As they are alternate interior angles )
So,
∠ CBE   =  ∠ CEB  ( As we know ∠ CBE  = ∠ EBA )
So,
BC  = EC( As we know base angle theorem , if base angles of any triangle is equal than opposite sides are also equal )
Hence
AD  =  DE  =  BC  =  EC   ---- ( 1 )
We know
AB  =  CD
And
AB  =  DE  +  EC   ( As CD  =  DE  +  EC )