In the diagram below, triangle \(ABC\) is an equilateral triangle, and triangle \(DEF\) is an isosceles triangle where \(DE = DF\). The measure of \(\angle DEF = 40^\circ\). What is the measure of \(\angle ADF\) in degrees?
[asy]
size(200);
pair A = (0,0), B = (3,0), C = (1.5,2.6), D = (1.5,0), E = (0.5,-1.5), F = (2.5,-1.5);
draw(A--B--C--cycle);
draw(D--E--F--cycle);
label("$A$", A, SW);
label("$B$", B, SE);
label("$C$", C, N);
label("$D$", D, S);
label("$E$", E, SW);
label("$F$", F, SE);
label("$40^\circ$", (1.5,-1.5), N);
\end{asy} Let's think step by step and output the final answer within \boxed{}.
user