https://file.aiquickdraw.com/mathbot/file/1726092994772j2siidsk.jpg

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:

| **Statement**                                   | **Reason**                                      |
|-------------------------------------------------|-------------------------------------------------|
| 1. ∠1 and ∠2 form a linear pair                 | 1. Given (Linear Pair Theorem)                  |
| 2. ∠1 and ∠2 are supplementary                  | 2. Linear Pair Theorem (angles in a linear pair are supplementary) |
| 3. ∠1 ≅ ∠3                                      | 3. Given (Angle congruence)                     |
| 4. ∠2 and ∠3 are supplementary                  | 4. Substitution property (since ∠1 ≅ ∠3)        |

### 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.

Identify the correct two-column proof for this plan. Given: ∠1 ≅ ∠3. Prove: ∠2 and ∠3 are supplementary.

Learn how to prove that ∠2 and ∠3 are supplementary using the Linear Pair Theorem and angle congruence. This step-by-step two-column proof covers key geometric concepts such as congruence, supplementary angles, and substitution. Suitable for students in Grades 8-10.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation