Let's go through the two-column proof from the image. The problem asks to prove that angles ∠1 and ∠4 are supplementary, with the following given information:

Given:

• $a \parallel b$
• $c \parallel d$

Prove:

• ∠1 and ∠4 are supplementary.

Two-Column Proof Layout:

Here’s the given table of statements and reasons, where we need to fill in the missing reasons:

Completing the Proof:

1. Statement 1 (∠1 ≅ ∠2):

   • Reason: Alternate Interior Angles Theorem (Since $c \parallel d$, ∠1 and ∠2 are alternate interior angles, so they are congruent.)

2. Statement 3 (∠2 and ∠3 are supplementary):

   • Reason: Linear Pair Postulate (Since ∠2 and ∠3 form a straight line, they are a linear pair and are supplementary.)

3. Statement 5 (∠3 ≅ ∠4):

   • Reason: Alternate Interior Angles Theorem (Since $a \parallel b$, ∠3 and ∠4 are alternate interior angles, so they are congruent.)

4. Statement 6 (∠1 and ∠4 are supplementary):

   • Reason: Congruent Supplements Theorem (If two angles are supplementary to the same angle or congruent angles, then they are supplementary to each other.)

Final Table:

Would you like a deeper explanation of any of the steps?

Here are 5 related questions to explore further:

1. What is the difference between alternate interior angles and corresponding angles?
2. Can supplementary angles always be adjacent?
3. How does the Congruent Supplements Theorem apply in other scenarios?
4. Why do parallel lines play a critical role in angle congruency theorems?
5. How can you prove that two angles are congruent without using alternate interior angles?

Tip: Remember that alternate interior angles are congruent only when the lines are parallel.