Problem Analysis:

We are tasked with proving that $\triangle BAE \cong \triangle CAD$ based on the given information:

1. $\overline{AD} \cong \overline{AE}$
2. $\triangle CFE \cong \triangle BFD$

Key Observations:

• Since $\triangle CFE \cong \triangle BFD$, we can deduce that corresponding sides and angles of these triangles are congruent.
• The problem prohibits using quadrilateral properties, so the proof must rely solely on triangle congruence criteria (e.g., SAS, ASA, SSS, etc.).

Step-by-Step Proof:

1. Start with the given:

   • $\overline{AD} \cong \overline{AE}$ (Given)
   • $\triangle CFE \cong \triangle BFD$ (Given)

2. Corresponding parts of congruent triangles: From $\triangle CFE \cong \triangle BFD$, the following corresponding parts are congruent:

   • $\overline{CF} \cong \overline{BF}$
   • $\overline{FE} \cong \overline{FD}$
   • $\angle CFE \cong \angle BFD$

3. Vertical angles: $\angle BFA \cong \angle CFA$ because they are vertical angles.

4. Angle-Side-Angle (ASA) congruence: Considering $\triangle BAE$ and $\triangle CAD$:

   • $\overline{AD} \cong \overline{AE}$ (Given)
   • $\angle BFA \cong \angle CFA$ (Vertical angles)
   • $\triangle CFE \cong \triangle BFD$ implies $\angle BAE \cong \angle CAD$ because these are corresponding angles.

   Therefore, $\triangle BAE \cong \triangle CAD$ by the ASA congruence criterion.

Conclusion:

We have proven $\triangle BAE \cong \triangle CAD$ using the given information, vertical angles, and the ASA criterion.

Do you need further explanation on any of the steps? Here are some related questions to explore:

1. How do vertical angles help in geometric proofs?
2. What are the properties of corresponding parts of congruent triangles (CPCTC)?
3. Why is ASA a valid congruence criterion?
4. How can congruent triangles be used to deduce equal segments or angles in a figure?
5. What other congruence criteria could apply in this situation (e.g., SAS or SSS)?

Tip: Always look for congruent triangles when solving geometric proofs—they often provide the key relationships needed!