1. **State the problem:** Given that E is the midpoint of segment AD, AB is perpendicular to AD, DC is perpendicular to AD, and angles \(\angle CBE \cong \angle BCE\), prove that segments AB and DC are congruent. 2. **Identify given information:** - E is midpoint of AD \(\Rightarrow AE = ED\) - \(AB \perp AD\) - \(DC \perp AD\) - \(\angle CBE \cong \angle BCE\) 3. **Goal:** Prove \(AB \cong DC\). 4. **Use properties of perpendicular lines:** Since both AB and DC are perpendicular to AD, AB and DC are parallel and form right angles with AD. 5. **Consider triangles \(\triangle ABE\) and \(\triangle DCE\):** - \(AE = ED\) because E is midpoint. - \(\angle AEB = \angle DEB = 90^\circ\) because AB and DC are perpendicular to AD. - \(\angle CBE \cong \angle BCE\) is given. 6. **Use the Angle-Side-Angle (ASA) congruence criterion:** - Side: \(AE = ED\) - Angle: \(\angle AEB = \angle DEB = 90^\circ\) - Angle: \(\angle CBE \cong \angle BCE\) 7. **Conclude:** By ASA, \(\triangle ABE \cong \triangle DCE\), so corresponding sides \(AB \cong DC\). **Final answer:** \(AB \cong DC\).