Answer
$m\angle DOE=48^{\circ}$
Work Step by Step
The measure of an entire circle equals $360^{\circ}$. Subtract out the major arc $ECABD$, or $312^{\circ}$.
$360-312=48$
Minor arc $DE=48^{\circ}$.
The measure of a central angle is equal to its intercepted arc. Thus,
$m\angle DOE=48^{\circ}$.
