Point p falls on point q and connects QP.
So BQ=BP=2, AQ=PC=3.
Because CBP angle = ABQ angle, QBP angle =90 degrees.
So QP=2* root number 2, angle QPB=45 degrees.
In the triangle APQ, AP= 1, AQ=3, QP=2 and root number 2.
That is, AP Square +QP Square =AQ Square (not difficult to see)
So APQ angle =90 degrees.
So angle APB= angle APQ+ angle QPB=90+45= 135 degrees.
The final answer is 135 degrees.