1. **State the problem:** Given that triangles $\triangle ADB$ and $\triangle BCA$ are congruent, prove that segments $DE$ and $CE$ are congruent. 2. **Given:** $\triangle ADB \cong \triangle BCA$. 3. **Identify corresponding parts:** Since the triangles are congruent, all corresponding sides and angles are congruent. Therefore, $AD \cong BC$, $DB \cong CA$, and $\angle D \cong \angle C$. 4. **Locate point $E$:** Point $E$ lies on segments $DB$ and $AC$. Since $DB \cong AC$, and $E$ is a point on both, segments $DE$ and $CE$ are parts of these congruent segments. 5. **Use the property of congruent segments:** Because $DB \cong AC$, and $E$ is a point dividing these segments, the segments $DE$ and $CE$ correspond to each other. 6. **Conclude:** Therefore, $DE \cong CE$. **Final answer:** $DE \cong CE$.