In parallelogram $ABCD$, the measure of angle $ABC$ is 3 times the measure of angle $BCD$. How many degrees are in the measure of angle $ADC$?
[asy]
unitsize(1inch);
pair A,B,C,D;
B = (0,0);
C = (1,0);
A = rotate(135)*(0.6,0);
D = A+C;
draw(A--B--C--D--A);
label("$A$",A,N);
label("$D$",D,N);
label("$B$",B,S);
label("$C$",C,S);
[/asy]

Because $\overline{AB}\parallel\overline{CD}$, we have $\angle B + \angle C = 180^\circ$.  Since $\angle B = 3\angle C$, we have $3\angle C + \angle C = 180^\circ$, so $4\angle C = 180^\circ$ and $\angle C = 45^\circ$.  Since $\overline{AD}\parallel\overline{BC}$, we have $\angle C +\angle D = 180^\circ$, so $\angle D = 180^\circ - \angle C = \boxed{135^\circ}$.