In the figure below, \(\overleftrightarrow{AB}\) and \(\overleftrightarrow{CE}\) intersects at \(O\), \(\overrightarrow{OC}\) bisects \(\angle{BOD}\), and the measure of \(\angle{AOD}\) is \(40^\circ\). What is the measure of \(\angle{AOE}\) ?

Summary Submit Skip Question

Since \(\overrightarrow{OC}\) bisects \(\angle{BOD}\), \( \angle{DOC} \cong \angle{COB}\).

We can sum up all of the angles on the top half of the circle to \(180^\circ\).

$$ 40^\circ + m\angle{DOC} + m\angle{COB} =180^\circ$$ $$ 40^\circ + x + x = 180^\circ $$ $$ 2x = 140 $$ $$ x = 70^\circ $$

Since \(\angle{AOE}\) and \(\angle{COB}\) are vertical angles, they are congruent as well.