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^@#