# Answer the whole 17the question."DON'T SEND STHE SIMILAR QUERIES.KINDLY PERFORM THE STEP BY STEP AND APPROPRIATE ANSWER". Dear Student,

Please find below the solution to the asked query:

We form our diagram from given information , As : Here , PQ | | RS and XY is a transversal line .

i )  AC , AD is angle bisectors of $\angle$ PAB and $\angle$ QAB respectively and BC , BD is angle bisectors of $\angle$ RBA and $\angle$ SBA respectively . So

$\angle$ PAC =  $\angle$ CAB =   , So

2 $\angle$ PAC = 2 $\angle$ CAB = $\angle$ PAB                                       --- ( 1 )

And

$\angle$ PAD =  $\angle$ DAB =   , So

2 $\angle$ PAD = 2 $\angle$ DAB = $\angle$ QAB                                       --- ( 2 )

And

$\angle$ PAB +  $\angle$ QAB  =  180$°$                                                     ( Linear pair angles )

Substitute values from equation 1 and 2 and get  :

2 $\angle$ CAB +  2 $\angle$ DAB =  180$°$ ,

2 ( $\angle$ CAB +  $\angle$ DAB ) = 180$°$ ,

2 $\angle$ CAD =  180$°$ ,

$\angle$ CAD =  90$°$

Similarly we can show $\angle$ CBD = 90$°$  , So

$\angle$ CAD = 90$°$   and $\angle$ CBD = 90$°$                                               --- ( M )

And

$\angle$ PAB + $\angle$ RBA  = 180$°$                                         ( Consecutive interior angles , PQ | | RS and XY is transversal line )

And

2 $\angle$ CAB +  2 $\angle$ CBA = 180$°$                                   ( From equation 1 and as given BCis angle bisectors of $\angle$ RBA )

2 ( $\angle$ CAB +  $\angle$ CBA ) =  180$°$ ,

$\angle$ CAB +  $\angle$ CBA = 90$°$                                                 --- ( 3 )

And from angle sum property of triangle we get in triangle ABC :

$\angle$ CAB +  $\angle$ CBA + $\angle$ ACB =  180$°$ , Now we substitute value from equation 3 and get :

90$°$  + $\angle$ ACB =  180$°$ ,

$\angle$ ACB =  90$°$

Similarly we can get : $\angle$ BDA =  90$°$  , So

$\angle$ ACB = 90$°$   and $\angle$ BDA = 90$°$                                               --- ( N )

From equation ' M ' and ' N '  we can say that :

ACBD is a rectangle .                                                ( Hence proved )

Try to find solution for second part by yourself by considering above diagram and if you still have any more doubts just ask here on the forum and our experts will try to help you out as soon as possible.

Regards

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