In triangle $abc$, $ab=ac$ and $d$ is a point on $\overline{ac}$ so that $\overline{bd}$ bisects angle $abc$. if $bd=bc$, what is the measure, in degrees, of angle $a$?

Angle a is 36 degrees. Suppose angle abd is x degrees, then angle cbd is also x, since bd is the angle bisector of angle abc, which is equal to 2x. Since bd=bc, angle bdc=angle bcd=(180-x)/2. Since ab=ac, angle abc is equal to angle bcd. Therefore, 2x=(180-x)/2, x=36. Angle abc=bcd=72 degrees, so angle a=180-72*2=36 degrees.