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 $\angle{AOE}$ ?

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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.