AMC 8 1996 Problem 24

The measure of angle $ABC$ is $50^\circ$, $\overline{AD}$ bisects angle $BAC$, and $\overline{DC}$ bisects angle $BCA$. The measure of angle $ADC$ is

$\text{(A)}\ 90^\circ \text{(B)}\ 100^\circ \text{(C)}\ 115^\circ \text{(D)}\ 122.5^\circ \text{(E)}\ 125^\circ$