To prove that $\triangle ABD \cong \triangle CDB$, we will use the triangle congruence properties systematically, step by step. Here’s how:

Step 1: Analyze the Given Information

1. It is given that:
   • $AB \parallel CD$
   • $AD \parallel BC$
2. The figure is a parallelogram because opposite sides are parallel.

Step 2: Identify Relationships Between Triangles $\triangle ABD$ and $\triangle CDB$

1. Opposite sides of a parallelogram are equal:

   • $AB = CD$ (Given $AB \parallel CD$ implies $AB$ and $CD$ are equal in length.)
   • $AD = BC$ (Similarly, $AD \parallel BC$ implies $AD$ and $BC$ are equal in length.)

2. Diagonal $BD$ is common to both triangles:

   • $BD = BD$ (Shared side between $\triangle ABD$ and $\triangle CDB$.)

Step 3: Prove Congruence

Using the Side-Side-Side (SSS) Congruence Criterion, we can show:

1. $AB = CD$ (Corresponding sides in $\triangle ABD$ and $\triangle CDB$.)
2. $AD = BC$ (Corresponding sides in $\triangle ABD$ and $\triangle CDB$.)
3. $BD = BD$ (The common side.)

Thus, $\triangle ABD \cong \triangle CDB$ by the SSS Congruence Theorem.

Step 4: Conclusion

Since all three corresponding sides of $\triangle ABD$ and $\triangle CDB$ are equal, the triangles are congruent.

Reasoning:

1. The SSS Congruence Criterion requires that all three sides of one triangle are equal to the corresponding sides of another triangle.
2. Parallelogram properties (opposite sides are equal and parallel) ensure the conditions needed to apply the theorem.

Do you want me to illustrate this further with additional diagrams or alternative proofs? Here are some follow-up questions for deeper understanding:

1. Can you identify which other congruence theorems could work here (e.g., ASA, SAS)?
2. Why does the shared diagonal $BD$ play a crucial role in proving congruence?
3. How can the properties of a parallelogram help in proving triangle relationships?
4. What is the significance of parallel lines in establishing side equality?
5. How could this proof change if the shape was not a parallelogram?

Tip: Always start with given information and properties of geometric shapes when proving congruence.