1. **State the problem:** Given that $\overline{CA} \cong \overline{CE}$ and $\overline{BA} \cong \overline{DE}$, prove that $\overline{BX} \cong \overline{DX}$. 2. **Identify known information:** We have two pairs of congruent segments: $\overline{CA} \cong \overline{CE}$ and $\overline{BA} \cong \overline{DE}$. Also, vertical angles formed at point $X$ are congruent. 3. **Use triangle congruence criteria:** To prove $\overline{BX} \cong \overline{DX}$, consider triangles $\triangle BXA$ and $\triangle DXE$. We want to show these triangles are congruent. 4. **List congruent parts:** - $\overline{BA} \cong \overline{DE}$ (given) - $\overline{CA} \cong \overline{CE}$ (given) - $\angle BXA \cong \angle DXE$ (vertical angles are congruent) 5. **Apply SAS (Side-Angle-Side) congruence:** Since two sides and the included angle of $\triangle BXA$ and $\triangle DXE$ are congruent, by SAS, $\triangle BXA \cong \triangle DXE$. 6. **Conclude corresponding parts are congruent:** From the congruence of triangles, corresponding sides $\overline{BX} \cong \overline{DX}$. **Final answer:** $$\overline{BX} \cong \overline{DX}$$