Final answer:
Using the Angle-Angle-Side (AAS) Congruence Postulate, it's shown that triangles ADB and CDB are congruent, which, by CPCTC, proves that AD is congruent to CD.
Step-by-step explanation:
To prove that AD ≅ CD given that AB ≅ CB, ∠A ≅ ∠C, and DB bisects ∠ABC, we can use the properties of triangles and the congruency postulates. Since DB is a bisector, we know that ∠ADB ≅ ∠CDB. By the given, AB ≅ CB and ∠A ≅ ∠C. We now have two angles and a side that are congruent for both triangles ADB and CDB which suggests that we can use the Angle-Angle-Side (AAS) Congruence Postulate to show that triangles ADB and CDB are congruent. As a result, by the corresponding parts of congruent triangles are congruent (CPCTC), we have that AD ≅ CD.