drawn
Let /_QCD=alpha
/_PCB=beta
So /_PCQ=90^@-(alpha+beta)
Now for Delta QCD,/_QDC=90^@ and DC = 1unit
So DQ=DCtanalpha=tanalpha
And for Delta PCB,/_PBC=90^@ and BC = 1unit
So PB=BCtanbeta=tanbeta
Hence for DeltaAPQ
AQ=1-tanalpha
AP=1-tanbeta
and
PQ=sqrt((1-tanalpha)^2+(1-tanbeta)^2)
Given
AQ+AP+PQ=2
=>1-tanalpha+1-tanbeta+sqrt((1-tanalpha)^2+(1-tanbeta)^2)=2
=>sqrt((1-tanalpha)^2+(1-tanbeta)^2)=tanalpha +tanbeta
=>1-2tanalpha+tan^2alpha+1-2tanbeta+tan^2beta=tan^2alpha+tan^2beta+2tanalphatanbeta
=>1-tanalpha-tanbeta=tanalphatanbeta
=>tanalpha+tanbeta=1-tanalphatanbeta
=>(tanalpha+tanbeta)/(1-tanalphatanbeta)=1=tan45^@
=>tan(alpha+beta)=tan45^@
=>alpha +beta=45^@
Hence /_PCQ=90^@-(alpha+beta)=90^@-45^@=45^@