1. **State the problem:** Prove that $\angle ABD$ and $\angle EBC$ are complementary angles given the statements and reasons. 2. **Step 1:** Given $\overline{BD} \perp \overline{BC}$, by definition of perpendicular lines, $\angle DBC$ is a right angle. 3. **Step 2:** Since $\angle DBC$ is a right angle, its measure is $90^\circ$, so $m \angle DBC = 90^\circ$. 4. **Step 3:** By the Angle Addition Postulate, the measure of $\angle DBC$ is the sum of the measures of $\angle DBE$ and $\angle EBC$: $$m \angle DBE + m \angle EBC = m \angle DBC$$ 5. **Step 4:** Substitute $m \angle DBC = 90^\circ$ into the equation: $$m \angle DBE + m \angle EBC = 90^\circ$$ 6. **Step 5:** Given $\angle ABD \cong \angle DBE$, by definition of congruent angles, their measures are equal: $$m \angle ABD = m \angle DBE$$ 7. **Step 6:** Substitute $m \angle DBE$ with $m \angle ABD$ in the sum: $$m \angle ABD + m \angle EBC = 90^\circ$$ 8. **Step 7:** By definition, two angles whose measures add up to $90^\circ$ are complementary. 9. **Conclusion:** Therefore, $\angle ABD$ and $\angle EBC$ are complementary angles. This completes the proof.