PQR is a equilateral triangle and QRST is a square.prove that:-

  1. PT=PS
  2. angle SR=15 degree

Answer :

We have PQR is a equilateral triangle and  QRST is a square  ,

SO,

PQ  =  QR  =  PR    ( sides of equilateral triangle )
And
QR  =  RS  =  ST  =  TQ    ( Sides of square )

And
PRS  =  PRQ  +  QRS  =  60°  + 90°  =  150°  ---- ( 1 )   ( As we know all angle of equilateral triangle is 60° and all angles on a square is 90° )

And
PQT  =  PQR  +  RQT   =  60°  + 90°  =  150°  ---- ( 2 )  ( As we know all angle of equilateral triangle is 60° and all angles on a square is 90° )
From equation 1 and 2 , we get

PRS  =  PQT   ---------------- ( 3 )

1 )  Now in PRS and PQT 

PR  =  PQ  ( Sides of equilateral triangle )

PRS  =  PQT   ( From equation 3  )

And
RS  =  QT  ( Sides of square )

Hence,

PRS PQT  ( By SAS  rule  )

So,

PT  =  PS  ( BY CPCT )  ( Hence proved )

2 )  Show PSR  =  15°

Now in PRS
PR  =  PS  ,
So , By base angle theorem , we get
SPR  =  PSR

And we know angle sum properties   in PRS

SPR  +  PSR  + PRS  = 180° 

PSR  +  PSR  + 150°  = 180°  ( As we know PRS  = 150°  And SPR  =  PSR  )

2 PSR  = 180°  -  150° 

2 PSR  = 30° 

  PSR  = 15°  (  Hence proved  )

  • 8

In triangle PRS;

angle PRS = angle PRQ +angle QRS

angle PRQ = 60 (angles of a equilateral riangle are 60)

and angle QRS = 90 (all angles of a square are 90)

thus angle PRS = 90 +60 = 150

similarly angle PQT = 150 .......(i)

In triangle PRS, trianlge PQT;

PQ= PR (sides of a equilateral triangle are equal)

angle PRS = angle PQT (by i)

QT =RS (all sides of a square a equal)

thus triangle PRS is congruent to triangle PQT

thus PS = PT (by cpct)

PR=RS(PR= RQ - they are sides of a equilateral triangle and SR= RQ - they are sides of a square. Thus PR =RQ and SR = RQ So, SR=PR)

Thus angle PSR = angle RPS = x (say)

thus;

angle x + angle x + angle PRS = 180 (Angle sum property)

2x +150 = 180

2x = 180 - 150 = 30

x =30/2 =15

thus : angle PSR =15 degrees

  • 12

thankyou siddharth.

  • 4
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