In the diagram below, $B,$ $C,$ and $D$ are all on the same line, $\angle BAC=24^\circ$, and $AB=AC=CD$.

If $\angle ADC=x^\circ$, what is the value of $x$?

[asy]
pair a=(0,cos(pi/15)); pair b=(-sin(pi/15),0); pair c=-b; pair d=c+(1,0);
dot(a); dot(b); dot(c); dot(d);
draw(c--a--b--d--a);
draw((2*a+3*b)/5-0.05*(cos(pi/15),- sin(pi/15))--(2*a+3*b)/5+0.05*(cos( pi/15),-sin(pi/15)));
draw((2*a+3*c)/5-0.05*(cos(pi/15),s in(pi/15))--(2*a+3*c)/5+0.05*(cos(p i/15),sin(pi/15)));
draw((c+d)/2+(0,0.05)--(c+d)/2-(0,0 .05));
label(scale(0.75)*"$24^\circ$",a-(0 ,0.3),S);
label("$A$",a, N);
label("$B$",b, SSW);
label("$C$",c, S);
label("$D$",d, SSE);
label(scale(0.85)*"$x^\circ$",d+(-0 .17,0.06));
[/asy]