1. **State the problem:** Given that \(\overline{AB} \cong \overline{BC}\) and \(\angle ABE \cong \angle CBD\), prove that \(\triangle BED\) is an isosceles triangle. 2. **Recall the given information:** \[\overline{AB} \cong \overline{BC}, \quad \angle ABE \cong \angle CBD\] 3. **Use the property of isosceles triangles:** In a triangle, angles opposite congruent sides are congruent. Therefore, \(\angle A \cong \angle C\). 4. **Apply ASA (Angle-Side-Angle) congruence:** Triangles \(ABE\) and \(CBD\) are congruent by ASA because they have two pairs of congruent angles and the included side \(\overline{AB} \cong \overline{BC}\). 5. **Missing statement and reason:** \[\overline{BE} \cong \overline{BD}\] Reason: Corresponding parts of congruent triangles are congruent (CPCTC). 6. **Conclusion:** Since \(\overline{BE} \cong \overline{BD}\), \(\triangle BED\) has two congruent sides and is therefore isosceles. **Final answer:** \(\overline{BE} \cong \overline{BD}\) by CPCTC.