https://file.aiquickdraw.com/mathbot/file/1728948509634u02uh4yv.jpg

Based on the provided image, you are working on a proof involving angles and congruence. The problem states that:

- **Given:**
  - ∠1 and ∠4 form a linear pair.
  - ∠1 and ∠2 are supplementary.
- **To prove:** ∠3 ≅ ∠4.

The two-column proof chart is partially filled, and you need to complete the missing statements and reasons. Let’s go through the reasoning step-by-step:

1. **Statement 1:** ∠1 and ∠4 form a linear pair.
   - **Reason 1:** Given.

2. **Statement 2:** ∠1 and ∠4 are supplementary.
   - **Reason 2:** Linear Pair (Supplement) Theorem (if two angles form a linear pair, they are supplementary).

3. **Statement 3:** ∠1 and ∠2 are supplementary.
   - **Reason 3:** Given.

4. **Statement 4:** ∠2 ≅ ∠3.
   - **Reason 4:** Congruent Supplements Theorem (if two angles are supplementary to the same angle, they are congruent).

5. **Statement 5:** ∠3 ≅ ∠4.
   - **Reason 5:** Transitive Property (if ∠2 ≅ ∠3 and ∠2 ≅ ∠4, then ∠3 ≅ ∠4).

The completed two-column proof is as follows:

| Statements                        | Reasons                                   |
|------------------------------------|-------------------------------------------|
| 1. ∠1 and ∠4 form a linear pair.   | 1. Given                                  |
| 2. ∠1 and ∠4 are supplementary.    | 2. Linear Pair (Supplement) Theorem       |
| 3. ∠1 and ∠2 are supplementary.    | 3. Given                                  |
| 4. ∠2 ≅ ∠3.                        | 4. Congruent Supplements Theorem          |
| 5. ∠3 ≅ ∠4.                        | 5. Transitive Property                    |

Would you like a detailed explanation for any of these steps? 

Here are 5 related questions:
1. What is the definition of a linear pair?
2. How does the Congruent Supplements Theorem work in geometry?
3. Why do supplementary angles add up to 180°?
4. Can you explain the Transitive Property in geometry with other examples?
5. How are supplementary angles and congruent angles related?

**Tip:** In proofs, always ensure that your reasons are logically sound and directly follow from the previous steps or the given information.

Given: ∠1 and ∠4 form a linear pair; ∠1 and ∠2 are supplementary. Prove: ∠3 ≅ ∠4.

In this geometry problem, we prove that ∠3 is congruent to ∠4 using the linear pair theorem, the congruent supplements theorem, and the transitive property. This proof is suitable for students in Grades 8-10 and involves understanding the relationships between angles and their properties.

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