PQRS is a rectangle.  QM and SN are perpendiculars from Q and S on PR. 
a) is triangle QMR concurrent to triangle SNP?
b) state the pairs of matching parts needed to answer this.
c) is it true that QM= SN ?

Dear Student,

Please find below the solution to the asked query:

We form our diagram from given information , As :

a ) In $∆$ QMR and $∆$ SNP 

QR =  SP                                                            ( Opposite sides of given rectangle )

$\angle \mathrm{QMR}\hspace{0.17em} = \angle \mathrm{SNP} = 90° ( \mathrm{Given} : \mathrm{QM} \perp \mathrm{PR}\hspace{0.17em}\mathrm{and} \mathrm{SN} \perp \mathrm{PR} )$

And

$\angle$ QRM  = $\angle$ SPN                                            ( Alternate interior angles , As here QR | | SP and we take PR as transversal line )

So,

$∆$ QMR $\cong$ $∆$ SNP                                           ( By AAS rule )                                    ( Hence proved )

b )  As we can see to show $∆$ QMR and $∆$ SNP are congruent we show matching pairs :

i ) QR =  SP ,

ii ) $\angle$ QMR  = $\angle$ SNP 

And

iii ) $\angle$ QRM  = $\angle$ SPN  

c ) As we have proved $∆$ QMR and $∆$ SNP are congruent , So

QM  =  SN                                                 ( By CPCT )                                 ( Hence proved )


Hope this information will clear your doubts about topic.

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