Straightedge and compass - translating an angle

Given two lines that meet at an angle of $\alpha$ , and a point $P$ , how many moves does it take to construct two points $A,B$ such that $\angle APB=\alpha$ ?

Note: This is just to construct the two points; the construction of lines $AP$ and $BP$ should not be counted in your answer.

All terminology in this question is explained in the first note of my straightedge and compass set. More straightedge and compass constructions can be found there.