ABCD is a square & angle PQR=90

Answer : Given : ABCD is a square & angle PQR=90 . 

P,Q,R are 3 points on AB,BC & CD respectively.AP=CR.

To Prove :

1.PB = QC

2.PQ=QR

3.Angle QPR=45

 

Let as assume that angle QPR =a and angle QRP = b

 

Now , in triangle PQR and triangle QCR

angle PQR = angle QCR = 90

angle PRQ = angle RQC = b .....(1) { opposite angles}

angle QPR = angle QRC = a....(2) {exterior angles}

 

therefore the triangles are congruent by AAA property. 

=> QR= QC .....(3)

similarily triangle PQR is congruent to triangle PBQ  and as well congruent to triangle QCR

=> QC = QR = PB        {using 3}

also QR = PQ     {of triangle QCR and triangle PBQ } ......(4)

 

 

 

now eq 4 suggests that in triangle PQR PQ = QR 

therefore triangle PQR is issoceles triangle 

=> angle a = angle b .....(5)      {angles of the issoceles triangles                                                   are equal  }

 

Now in triangle PQR

90 + angle a+ angle b = 180   

angle a+ angle b = 180 - 90 = 90

2 angle a = 90                 {using eq5}

=> angle a = 45 degree 

i.e angle QPR = 45 degree

Hence proved