We are given that $\overline{DB}$ bisects $\angle ABC$ and $\angle A \cong \angle C$. We need to prove that $\triangle ABD \cong \triangle CBD$.

1. Problem Description

We are given that $\overline{DB}$ bisects $\angle ABC$ and $\angle A \cong \angle C$. We need to prove that $\triangle ABD \cong \triangle CBD$.

2. Solution Steps

Step 1:

$\overline{DB}$ bisects $\angle ABC$ and $\angle A \cong \angle C$.

Reason: Given.

Step 2:

$\angle ABD \cong \angle CBD$

Reason: Definition of angle bisector.

Step 3:

$\overline{BD} \cong \overline{BD}$

Reason: Reflexive Property of Congruence.

Step 4:

$\angle A \cong \angle C$

Reason: Given.

Step 5:

$\triangle ABD \cong \triangle CBD$

Reason: Angle-Angle-Side (AAS) Congruence Theorem.

3. Final Answer

$\triangle ABD \cong \triangle CBD$