This problem involves constructing a two-column proof to show that ∠2 and ∠3 are supplementary.

Given:

• ∠1 ≅ ∠3 (congruence of angles)

Prove:

• ∠2 and ∠3 are supplementary.

Plan:

1. Use the Linear Pair Theorem: This theorem states that if two angles form a linear pair, then they are supplementary (i.e., their sum is 180°).
2. Use congruence and substitution to finalize the proof.

Two-Column Proof:

Explanation:

1. From the figure, ∠1 and ∠2 are adjacent angles that form a straight line, so by the Linear Pair Theorem, they are supplementary.
2. Given that ∠1 is congruent to ∠3, we can substitute ∠3 for ∠1 in the equation stating that ∠1 and ∠2 are supplementary. Thus, it follows that ∠2 and ∠3 are supplementary.

Would you like to explore any more details about this proof?

Here are five related questions:

1. What are the conditions for angles to form a linear pair?
2. How does the substitution property work in geometric proofs?
3. Can you explain how the congruence of angles affects their supplementary property?
4. What other theorems are commonly used in angle proofs like this?
5. How can you identify congruent angles in different geometric figures?

Tip: When writing a two-column proof, always ensure that each statement has a corresponding and valid reason. This is key to a logically sound proof.