We are tasked with proving that triangles $\triangle BCG \cong \triangle FEG$, given the following:

• $BC \cong FE$ (equal lengths),
• $DG$ is an angle bisector.

Let’s go step by step:

Steps to Prove $\triangle BCG \cong \triangle FEG$:

1. Given Information:

   • $BC \cong FE$: This indicates that the sides $BC$ and $FE$ are equal.
   • $DG$ is the bisector of $\angle BGF$: This means $\angle BGD = \angle FGD$.

2. Common Side:

   • $BG = FG$: $BG$ and $FG$ are the same segment since they share a length.

3. Angles:

   • Since $DG$ bisects $\angle BGF$, we have: $\angle BGD \cong \angle FGD$

4. Congruence Criterion:

   • By the SAS (Side-Angle-Side) criterion:
     • $BC \cong FE$,
     • $BG \cong FG$,
     • $\angle BGD \cong \angle FGD$, We can conclude: $\triangle BCG \cong \triangle FEG$

Final Proof:

Thus, by the SAS Congruence Rule, $\triangle BCG \cong \triangle FEG$.

Would you like me to elaborate on any part of the explanation or the congruence criteria? Here are five related questions:

1. What is the SAS Congruence Criterion in detail?
2. How does an angle bisector contribute to congruence proofs?
3. Can we use other congruence rules (e.g., ASA) to prove this?
4. Why is the common side $BG \cong FG$ critical to the proof?
5. What other properties of congruent triangles can be derived once proved?

Tip: Always identify all given elements and check for congruence rules like SAS, ASA, SSS, or RHS.