Polygon $ABCDEF$ is a regular hexagon. What is the measure in degrees of angle $ABF$?
In triangle $ABF$, the two acute angles are equal since $AB=AF$. Also, the measure of $\angle A$ is $180^\circ(6-2)/6=120^\circ$.  Letting $x$ be the measure of $\angle ABF$, we have \[
120^\circ+x+x=180^\circ \implies x=\boxed{30}\text{ degrees}.
\] [asy]
size(5cm);
defaultpen(linewidth(0.7));
int i;
pair A=dir(0), B=dir(60), C=dir(120), D=dir(180), Ep=dir(240), F=dir(300);
pair[] dots = {A,B,C,D,Ep,F};
string[] alphabet={"$A$","$B$","$C$","$D$","$E$","$F$"};
dot(dots);
for(i=0;i<6;++i)
{

draw(dir(60*i)--dir(60*i+60));

label(alphabet[i],dots[i],dots[i]);
}
draw(A--B--F);
draw(anglemark(F,B,A));
draw(anglemark(A,F,B));
[/asy]