We are given that $\overleftrightarrow{CE}$ and $\overleftrightarrow{FH}$ are parallel lines, and $m\angle FGD = 56^\circ$. We need to determine $m\angle CDB$.

Step-by-step solution:

1. Identify corresponding angles: Since $\overleftrightarrow{CE} \parallel \overleftrightarrow{FH}$, and $\overleftrightarrow{BD}$ is a transversal, the angles $\angle FGD$ and $\angle CDB$ are corresponding angles.

2. Corresponding angles theorem: According to the corresponding angles theorem, corresponding angles formed by a transversal cutting through two parallel lines are congruent. Therefore: $m\angle CDB = m\angle FGD$

3. Substitute the given value: We are given that $m\angle FGD = 56^\circ$, so: $m\angle CDB = 56^\circ$

Thus, the measure of $\angle CDB$ is $56^\circ$.

Would you like further details, or have any questions about this explanation?

Here are 5 related questions you might consider:

1. What are alternate interior angles and how do they apply in similar problems?
2. How can we prove that two lines are parallel using angles formed by a transversal?
3. If $m\angle FGD$ had been $40^\circ$, what would $m\angle CDB$ be?
4. How do supplementary angles factor into problems involving parallel lines and transversals?
5. How could you identify whether two angles are vertically opposite in a diagram like this?

Tip: When working with parallel lines and a transversal, remember that both alternate interior angles and corresponding angles are congruent.