jmc

The given information tells us that $A=90^{\circ }-B$ and $A=kB$ for some $k\ge 1$. Therefore, we have $kB=90^{\circ }-B$. This simplifies to $(k+1)B=90^{\circ }$. $k+1$ can be any factor of $90$ except one, since $k+1\ge 2$. $90=2\cdot 3^2\cdot 5$ has $2\cdot 3\cdot 2=12$ factors, so there are 11 possible values of $k$. Each value of $k$ uniquely determines the value of $B$ and therefore the value of $A$, so there are $\boxed{11}$ possible measures for $A$.