The image you've provided shows a proof where the goal is to prove that $\overleftrightarrow{AB} \parallel \overleftrightarrow{CD}$, given that $\angle EPB \cong \angle EQD$. The proof structure includes six statements and reasons, with some missing steps. Let's fill in the blanks.

Proof Structure:

Given:
$\angle EPB \cong \angle EQD$
Prove:
$\overleftrightarrow{AB} \parallel \overleftrightarrow{CD}$

Steps:

1. Statement: $\angle EPB \cong \angle EQD$
   Reason: Given.

2. Statement: $m\angle EPB = m\angle EQD$
   Reason: Definition of congruent angles.

3. Statement: $m\angle EPB + m\angle BPQ = 180^\circ$
   Reason: Linear Pair Postulate.

4. Statement: $m\angle EQD + m\angle BPQ = 180^\circ$
   Reason: Linear Pair Postulate (applied to the other angle pair).

5. Statement: $\angle EQD$ and $\angle BPQ$ are supplementary.
   Reason: Angles that add up to 180 degrees are supplementary.

6. Statement: $\overleftrightarrow{AB} \parallel \overleftrightarrow{CD}$
   Reason: Definition of parallel lines (corresponding angles postulate).

Explanation:

• Since $\angle EPB \cong \angle EQD$, and they are corresponding angles formed by the transversal $PQ$, by the Corresponding Angles Postulate, if two corresponding angles are congruent, the lines $AB$ and $CD$ must be parallel.

Would you like a more detailed explanation on any step?

Here are 5 related questions that can expand this discussion:

1. What is the Corresponding Angles Postulate and why is it important in proving parallel lines?
2. How does the Linear Pair Postulate help in angle relationship proofs?
3. What are supplementary angles and how are they used in geometric proofs?
4. What are the key properties of parallel lines in Euclidean geometry?
5. Can alternate interior angles also be used to prove lines are parallel?

Tip:

In geometry proofs, always ensure that each statement has a valid justification based on definitions, postulates, or theorems.