Question

abcd is a parallelogram the bisectors of angle a and b meet at e which lies on dc prove that ad =1/2ab

Answer 1

:-ABCD is a parallelogram.The bisectors of angle A and angle B meet at E, which lies on DC. Prove that AD= 1/2 Ab

Answer : 

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 )

AB  =  AD  +  AD                                                 ( from equation 1  )

AB  =  2 AD  

AD  = 1/2 AB                                                         ( Hence proved )

Answer 2

Answer:

Step-by-step explanation:

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 )

AB  =  AD  +  AD                                                 ( from equation 1  )

AB  =  2 AD  

AD  = 1/2 AB                                                         ( Hence proved )

Thank u