In the figure, RS = QT and QS = RT. Prove that PQ = QR.
       

                                 R
                                                                      Q
                                       S               
                                                                 T
                       
 
 
                                                    P       i cannot post the fig. plz answer without fig. if the fig is neccessary then just connect the the points
 

Answer :
Given
RS = QT  And  QS = RT

In $∆$QTR  And $∆$ QSR
RS = QT              ( Given )
QS = RT               ( Given )
QR =  QR            ( Common )
So
$∆$QTR $\cong$$∆$ QSR    ( By  SSS rule )
$\Rightarrow$ $\angle$ QRS = $\angle$ RQT   ---------  ( 1 )    ( CPCT )
$\angle$ QRT  = $\angle$ RQS        ---------  ( 2 )
Now equation 2  -  equation 1 , we get
$\angle$ SRT  =  $\angle$ TQS       ---------- ( 3 )
Also
$\angle$ RSQ = $\angle$ QTR
Both side subtract from 180$°$ ,  we get
180$°$ -  $\angle$ RSQ = 180$°$  - ​$\angle$ QTR

$\angle$ QSP = $\angle$ RTP        ------------- ( 4 )
From equation 3 and 4 , we get from AA rule
$∆$RPT $\cong$$∆$ QPS          ( By AA rule )

$\therefore$  PQ  = PR             ( CPCT )      

( Hence proved )