Triangle $ABC$ has vertices $A(1,1)$, $B(1,5)$, and $C(5,1)$. Triangle $DEF$ is formed such that its vertices are $D(1,1)$, $E(1,1+4k)$, and $F(1+4k,1)$ for some positive constant $k$. If triangle $DEF$ is similar to triangle $ABC$, what is the measure, in degrees, of $\angle F$?