pa is perpendicular to ab , qb is perpendicular to ab .and pa=qb.prove triangle oap congruent to triangle obq.is oa =op

Answer :

From given information we form our diagram , as , and we take point  " O "  where line PB and QA intersect .


Now In PAB and QBA

PA  =  QB                                            ( Given )

PAB  =  QBA  = 90°              ( Given )
And
AB =  AB                                           (  Common side )

Hence

PAB QBA                          ( By SAS rule )
So,

APB  =  BQA                        (  By CPCT )                                                 --------------- ( 1 )

And
PB  =  QA                                          (  By CPCT )                                                 --------------- ( 2 )

Now In OAP and OBQ

PA  =  QB                                            ( Given )

APO  =  BQO                          (  From equation 1 ) (  Here APB   and  APO are same angles and  BQA  and  BQO    are same angles )
And
AOP  =  BOQ                         (  Vertically opposite angles )
Hence

OAP OBQ                          ( By AAS rule )                                              ( Hence proved )
So,

OA =  OB                                            (  By CPCT )                                                 --------------- ( 3 )

OP =  OQ                                            (  By CPCT )                                                 --------------- ( 4 )

So. from equation 2 ,3 and 4 , we get

OA  =  OP                                          ( Hence proved )

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