The problem is a geometric proof involving parallel lines and angle congruence. Let's break it down step by step.

Given:

• $a \parallel b$
• $\angle 1 \cong \angle 2$

Prove:

• $c \parallel d$

Proof:

1. Statement 1: $a \parallel b$
   Reason 1: Given.

2. Statement 2: $\angle 1 \cong \angle 3$
   Reason 2: Corresponding Angle Postulate (since $a \parallel b$, corresponding angles are congruent).

3. Statement 3: $\angle 1 \cong \angle 2$
   Reason 3: Given.

4. Statement 4: $\angle 2 \cong \angle 3$
   Reason 4: Transitive Property of Equality (If $\angle 1 \cong \angle 3$ and $\angle 1 \cong \angle 2$, then $\angle 2 \cong \angle 3$).

5. Statement 5: $c \parallel d$
   Reason 5: Converse of the Corresponding Angle Postulate (since $\angle 2 \cong \angle 3$, the lines $c$ and $d$ must be parallel).

The Complete Proof:

The logic follows from the properties of corresponding angles and the transitive property, allowing us to conclude that $c \parallel d$.

Would you like more details on any part of this proof? Let me know if you have any questions.

Here are five follow-up questions to consider:

1. What is the Corresponding Angle Postulate?
2. How does the Transitive Property of Equality work in geometry?
3. Why does the converse of the Corresponding Angle Postulate prove that lines are parallel?
4. What other ways can we prove lines are parallel using angles?
5. How would the proof change if $\angle 1$ were not congruent to $\angle 2$?

Tip: When working with parallel lines, always look for angle pairs (like corresponding or alternate interior angles) that can help in forming congruence relationships!