1. **Problem statement:** Given that BE and AD bisect each other, prove that segment AB is congruent to segment DE. 2. **Understanding the problem:** When two segments bisect each other, their intersection point divides each segment into two equal parts. 3. **Given:** BE and AD bisect each other at point C. 4. **To prove:** $AB \cong DE$ 5. **Step 1:** Since BE and AD bisect each other at C, we have: $$BC = CE \quad \text{and} \quad AC = CD$$ 6. **Step 2:** Consider triangles $ABC$ and $DEC$. 7. **Step 3:** In triangles $ABC$ and $DEC$: - $BC = CE$ (given by bisection) - $AC = CD$ (given by bisection) - $\angle BCA = \angle ECD$ (vertical angles are congruent) 8. **Step 4:** By the Side-Angle-Side (SAS) congruence postulate, triangles $ABC$ and $DEC$ are congruent: $$\triangle ABC \cong \triangle DEC$$ 9. **Step 5:** Corresponding parts of congruent triangles are congruent (CPCTC), so: $$AB \cong DE$$ 10. **Conclusion:** We have proved that $AB$ is congruent to $DE$ as required.