To determine which triangle congruence criterion proves $\triangle BEC \cong \triangle DEA$, let us analyze the given diagram and the information provided:

1. Given Information:

   • $\overline{BD}$ is a perpendicular bisector of $\overline{AC}$.
   • $\overline{AC}$ is bisected at point $E$, making $\overline{AE} \cong \overline{EC}$.
   • $\angle BEA$ and $\angle CED$ are right angles due to the perpendicular bisector $\overline{BD}$.
   • $\overline{BD}$ is common to both triangles $\triangle BEC$ and $\triangle DEA$.

2. Triangle Congruence Criterion:

   • In $\triangle BEC$ and $\triangle DEA$, the following congruent elements are observed:

     • Side: $\overline{AE} \cong \overline{EC}$ (from the bisection of $\overline{AC}$).
     • Angle: $\angle BEA \cong \angle CED$ (both are right angles).
     • Side: $\overline{BD} \cong \overline{BD}$ (common side).

   • This corresponds to the Side-Angle-Side (SAS) triangle congruence criterion.

Final Answer:

The triangle congruence that proves $\triangle BEC \cong \triangle DEA$ is Side-Angle-Side (SAS).

Would you like further clarification or details about this reasoning? Here are some related questions to deepen your understanding:

1. What other triangle congruence criteria exist, and how do they differ from SAS?
2. Why is $\angle BEA$ and $\angle CED$ being right angles important in proving congruence?
3. Could $\triangle BEC$ and $\triangle DEA$ be congruent by Hypotenuse-Leg? Why or why not?
4. What role does the perpendicular bisector play in this proof?
5. How can we extend this reasoning to other geometrical proofs involving perpendicular bisectors?

Tip: Always look for shared sides or angles when comparing triangles—they are crucial for establishing congruence!