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In the figure, if PB=QC=DR, then we can deduce that DR = QR. This means PQ = QR and the triangle PQR is right as there is a 90degree angle. Since the two sides are equal, they are isosceles. So the angles PQR and QRP must be the same. Since three angles' sum is 180, the angles are of measure (180 - 90) / 2 = 45degrees.

The line BQC is a straight line. So angle BQP+ angle PQR + angle CQR =180 degrees. and angle BQP+ angle PQR +angle BPQ=180 degrees. NOW [angle BQP+angle BPQ] +angle PQR=180 degrees. 180-angle PBQ +angle PQR=180 degrees.[due to angle sum property] 180-90+angle PQR=180 degrees angle PQR=90 degrees so it is proved that traingle PQR is an isosceles triangle. and thats why angle QPR =45 degrees. hence proved..