Perpendicular Chord Bisector Theorem

Consider the segments $\overline{OC}$ and $\overline{OD}.$ Since $\overline{AB}$ and $\overline{CD}$ are perpendicular segments, then $\angle OEC$ and $\angle OED$ are right angles. Therefore, $△OCE$ and $△ODE$ are right triangles.

Next, since two radii of a circle are congruent, then $\overline{OC}$ and $\overline{OD}$ are congruent. Furthermore, by the Reflexive Property of Congruence, $\overline{OE}$ is congruent to itself. Therefore, the hypotenuse and one leg of $△OCE$ are congruent to the hypotenuse and the corresponding leg of $△ODE.$ By the Hypotenuse Leg Theorem, $△OCE$ and $△ODE$ are congruent triangles. Since corresponding parts of congruent triangles are congruent, then the other legs of the triangles are also congruent.

The proof for the first part of the statement has been completed.

Now, to show the congruence of the arcs $\overgroup{BC}$ and $\overgroup{BD},$ the properties of congruent right triangles will be considered. Since corresponding parts of congruent triangles are congruent, it can be said that $\angle COE$ and $\angle DOE$ are congruent angles. Therefore, by the Congruent Central Angles Theorem, $\overgroup{BC}$ and $\overgroup{BD}$ are congruent.

This completes the proof.