A circle is centered at $O$, $\overline{AB}$ is a diameter and $C$ is a point on the circle with $\angle COB = 50^\circ$.  What is the degree measure of $\angle CAB$?
Note that $\angle AOC = 180^\circ - 50^\circ = 130^\circ$.  Because triangle $AOC$ is isosceles, $\angle CAB = (180^\circ - 130^\circ)/2 = \boxed{25^\circ}$.

[asy]
import graph;

unitsize(2 cm);

pair O, A, B, C;

O = (0,0);
A = (-1,0);
B = (1,0);
C = dir(50);

draw(Circle(O,1));
draw(B--A--C--O);

label("$A$", A, W);
label("$B$", B, E);
label("$C$", C, NE);
label("$O$", O, S);
[/asy]