I can see the problem is about proving that $\triangle ACF \cong \triangle ECF$ based on the given information and using a two-column proof format. Let me extract and analyze the content.

Given:

1. $AB \cong DE$
2. $\triangle HBC \cong \triangle IDC$
3. $FG$ bisects $\angle HGI$

To Prove:

$\triangle ACF \cong \triangle ECF$

Proof Strategy:

We will use the Angle-Side-Angle (ASA) or Side-Angle-Side (SAS) congruence criteria to prove the two triangles are congruent. Here's how the reasoning will proceed:

Steps and Reasoning:

1. Given Information:

   • $AB \cong DE$
   • $\triangle HBC \cong \triangle IDC$
   • $FG$ bisects $\angle HGI$ (from the problem statement).

2. Shared Side:

   • $CF$ is a shared side between $\triangle ACF$ and $\triangle ECF$ (Reflexive Property).

3. Bisected Angle:

   • Since $FG$ bisects $\angle HGI$, we know that $\angle HCF \cong \angle ICF$.

4. Angle-Side-Angle (ASA):

   • $\angle HCF \cong \angle ICF$ (from Step 3).
   • $CF \cong CF$ (shared side, Reflexive Property).
   • $AB \cong DE$ implies that the distances from $A$ to $C$ and $E$ to $C$ maintain symmetry.
   • Therefore, $\triangle ACF \cong \triangle ECF$ by ASA.

Would you like me to format this into a full two-column proof or clarify any specific steps? Let me know if you have any questions or need further details.

Expanding Your Understanding:

Here are some follow-up questions related to this proof:

1. Why does the Reflexive Property apply to $CF$ in this case?
2. How does the congruence $\triangle HBC \cong \triangle IDC$ contribute to proving the main statement?
3. What does it mean geometrically for $FG$ to bisect $\angle HGI$?
4. Could $\triangle ACF \cong \triangle ECF$ also be proven using another congruence criterion (e.g., SAS)?
5. Why is it important that $AB \cong DE$ is explicitly stated in the problem?

Tip:

Always start a proof by identifying shared sides or angles—these often simplify congruence proofs significantly.