This question asks you to prove that angles ∠3 and ∠4 are supplementary given certain conditions about parallel lines and congruent angles. Let's go through the proof and fill in each statement and reason.

Given Information

1. $c \parallel d$
2. $\angle 1 \cong \angle 2$

Proving Steps

To prove: ∠3 and ∠4 are supplementary.

Step-by-Step Solution:

1. Statement: $c \parallel d$
   Reason: Given

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

3. Statement: $\angle 1 \cong \angle 3$
   Reason: Corresponding Angles Postulate (Since $a \parallel b$, angle 1 and angle 3 are corresponding angles.)

4. Statement: $\angle 2 \cong \angle 3$
   Reason: Transitive Property of Congruence (Since $\angle 1 \cong \angle 2$ and $\angle 1 \cong \angle 3$, by transitivity, $\angle 2 \cong \angle 3$.)

5. Statement: $a \parallel b$
   Reason: Corresponding Angles Converse Postulate (If $\angle 2 \cong \angle 3$, then lines $a$ and $b$ must be parallel.)

6. Statement: $\angle 3$ and $\angle 4$ are supplementary
   Reason: Same-Side Interior Angles Postulate (Since $a \parallel b$, angles 3 and 4 are supplementary.)

Would you like a more detailed explanation on any of these steps?

Related Questions:

1. What is the Corresponding Angles Postulate, and how is it applied in geometry proofs?
2. How does the Transitive Property of Congruence help in angle congruency proofs?
3. What is the Same-Side Interior Angles Postulate, and when can it be used?
4. How do parallel lines affect the relationships between angles formed by a transversal?
5. Why is it important to prove that lines are parallel before applying certain angle relationships?

Tip: When working with parallel lines and transversals, always look for patterns like corresponding, alternate interior, and same-side interior angles—they’re essential in geometry proofs.