This is a triangle proof problem involving congruent segments and the use of reasoning to prove that $\overline{DB}$ bisects $\angle ABC$. Here's how the proof can be structured step by step:

Given:

1. $\overline{AB} \cong \overline{CB}$
2. $\overline{AD} \cong \overline{CD}$

To Prove:

$\overline{DB}$ bisects $\angle ABC$.

Steps for Proof:

1. Statement: $\overline{AB} \cong \overline{CB}$
   Reason: Given.

2. Statement: $\overline{AD} \cong \overline{CD}$
   Reason: Given.

3. Statement: $\overline{BD} \cong \overline{BD}$
   Reason: Reflexive property of congruence.

4. Statement: $\triangle ABD \cong \triangle CBD$
   Reason: SSS (Side-Side-Side) congruence postulate.

5. Statement: $\angle ABD \cong \angle CBD$
   Reason: Corresponding parts of congruent triangles (CPCTC).

6. Statement: $\overline{DB}$ bisects $\angle ABC$
   Reason: Definition of an angle bisector (if $\angle ABD \cong \angle CBD$, $\overline{DB}$ bisects $\angle ABC$).

Would you like further details or explanations on any specific step? Here are some related questions to expand your understanding:

1. Why is the reflexive property essential in triangle congruence proofs?
2. How does SSS congruence differ from other triangle congruence postulates?
3. What are the key indicators that a line segment bisects an angle?
4. Why does CPCTC work for proving parts of triangles are congruent?
5. Can you use a different triangle congruence postulate to prove this statement?

Tip: Always start with the given information and look for congruent triangles as a way to unlock the proof.