if two lines re-intersected by a transversal line and if any pair of bisectors of their alternate interior angles is parallel then prove that two lines are parallel

Answer :

From given information we form our diagram , As  :

Here two lines AB and CD intersected by transversal line " XY "  and EG and FH are angle bisectors of $\angle$ AEF and $\angle$ DFE respectively .
So,

$\angle$ AEG  =  $\angle$ FEG  = $\frac{\angle \mathrm{AEF}}{2}$                              ------ ( 1 )

And


$\angle$ DFH  =  $\angle$ EFH  = $\frac{\angle \mathrm{DFE}}{2}$                              ------ ( 2 )

And also given , EG  | | FH ( bisectors of their alternate interior angles is parallel )

So,

$\angle$ FEG  =  $\angle$ EFH                                            ------ ( 3 )                  (  Alternate interior angles as EG  | | FH and XY is transversal line )

From equation 1 , 2 and 3 we get

$\angle$ AEF  =  $\angle$ DFE 

And that are alternate interior angles of line AB and CD and XY as transversal line only be true for AB | | CD                        ( Hence proved )