In the figure, lines $\overline{AB}$ and $\overline{CD}$ are parallel. Triangle $ABC$ is formed such that $\angle CAB = 55^\circ$ and $\angle ABC = 70^\circ$. What is the measure of $\angle BCD$?

[asy]
pair A, B, C, D;
A = (0,0);
B = (2,0);
C = (1,1.5);
D = (3,1.5);
draw(A--B--C--cycle);
draw(C--D);
draw(A--D);
label("$A$", A, SW);
label("$B$", B, SE);
label("$C$", C, N);
label("$D$", D, NE);
markangle(B, A, C);
markangle(A, B, C);
label("$55^\circ$", A, NE, rotate(55,A));
label("$70^\circ$", B, NW, rotate(70,B));
label("$\overline{AB} \parallel \overline{CD}$", (1, -0.5));
\end{asy} Let's think step by step and output the final answer within \boxed{}.
user