1. **State the problem:** Given that line segments $\overline{BD}$ and $\overline{AC}$ bisect each other at point $E$, prove that $\overline{AB} \cong \overline{CD}$ without using quadrilateral properties. 2. **Understand the given:** Since $\overline{BD}$ and $\overline{AC}$ bisect each other, point $E$ is the midpoint of both $\overline{BD}$ and $\overline{AC}$. This means: $$ BE = ED \quad \text{and} \quad AE = EC $$ 3. **Use midpoint definition:** By definition of midpoint, the segments on either side of $E$ are equal: $$ BE = ED \quad \Rightarrow \quad BE \cong ED $$ $$ AE = EC \quad \Rightarrow \quad AE \cong EC $$ 4. **Consider triangles:** Look at triangles $\triangle ABE$ and $\triangle CDE$. 5. **Identify congruent parts:** - $AE \cong CE$ (from midpoint of $\overline{AC}$) - $BE \cong DE$ (from midpoint of $\overline{BD}$) - $\angle AEB \cong \angle CED$ (vertical angles are congruent) 6. **Apply SAS congruence:** Triangles $\triangle ABE$ and $\triangle CDE$ have two sides and the included angle congruent, so: $$ \triangle ABE \cong \triangle CDE $$ 7. **Conclude segment congruence:** Corresponding parts of congruent triangles are congruent (CPCTC), so: $$ \overline{AB} \cong \overline{CD} $$ **Final answer:** $\overline{AB} \cong \overline{CD}$ as required.