Yes, I can help you with this! Here's how to prove $\triangle ABD \cong \triangle CDB$ step-by-step with reasoning:

Given:

1. $\overline{AB} \parallel \overline{CD}$
2. $\overline{AD} \parallel \overline{BC}$

To Prove:

$\triangle ABD \cong \triangle CDB$

Proof:

Step 1: Identify given parallel sides and transversal.

• $\overline{AB} \parallel \overline{CD}$ and $\overline{AD} \parallel \overline{BC}$.
• The diagonal $\overline{BD}$ acts as a transversal to both sets of parallel lines.

Step 2: Establish corresponding angles.

1. Since $\overline{AB} \parallel \overline{CD}$ and $\overline{BD}$ is a transversal:

   • $\angle ABD = \angle CDB$ (Alternate interior angles are equal).

2. Similarly, since $\overline{AD} \parallel \overline{BC}$ and $\overline{BD}$ is a transversal:

   • $\angle ADB = \angle CBD$ (Alternate interior angles are equal).

Step 3: Common side.

• $\overline{BD}$ is common to both $\triangle ABD$ and $\triangle CDB$.

Step 4: Apply ASA (Angle-Side-Angle) congruence criterion.

• In $\triangle ABD$ and $\triangle CDB$:
  1. $\angle ABD = \angle CDB$ (Step 2).
  2. $\overline{BD}$ is the common side (Step 3).
  3. $\angle ADB = \angle CBD$ (Step 2).

Thus, $\triangle ABD \cong \triangle CDB$ by the ASA criterion.

Conclusion:

$\triangle ABD \cong \triangle CDB$, as proven by the ASA congruence criterion.

Do you have questions or need further clarification?

Here are 5 related questions for practice:

1. How does the property of parallel lines help in proving triangle congruence?
2. Can $\triangle ABD \cong \triangle CDB$ be proven using other congruence criteria?
3. Why is the diagonal $\overline{BD}$ considered a transversal?
4. What other properties can be inferred if two triangles are congruent?
5. Can this proof work if one pair of sides was not parallel?

Tip: When proving triangle congruence, always identify the shared sides or angles first to simplify the process.