1. **State the problem:** Prove that angles \(\angle ABD\) and \(\angle EBC\) are complementary given the perpendicularity and angle relationships in the figure. 2. **Given:** \(BD \perp BC\) means \(\angle DBC\) is a right angle. 3. **Recall:** A right angle measures 90°. 4. From the given, \(m\angle DBC = 90^\circ\). 5. By the Angle Addition Postulate, \(m\angle DBC = m\angle DBE + m\angle EBC\). 6. Substitute the right angle measure: \(m\angle DBE + m\angle EBC = 90^\circ\). 7. Given \(\angle ABD \cong \angle DBE\), so their measures are equal: \(m\angle ABD = m\angle DBE\). 8. Substitute \(m\angle ABD\) for \(m\angle DBE\) in the sum: \(m\angle ABD + m\angle EBC = 90^\circ\). 9. By definition, two angles whose measures add up to 90° are complementary. **Final conclusion:** \(\angle ABD\) and \(\angle EBC\) are complementary angles.