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To construct a two-column proof of the **Alternate Exterior Angles Theorem**, we need to prove that \(\angle 1 \cong \angle 2\) given that lines \(m\) and \(n\) are parallel.

### Step-by-step guide:

- **Given**: \(m \parallel n\) (Lines \(m\) and \(n\) are parallel)
- **Prove**: \(\angle 1 \cong \angle 2\) (Angles 1 and 2 are congruent)

### Two-column proof:

| **Statements**                          | **Reasons**                                 |
|-----------------------------------------|---------------------------------------------|
| 1. \(m \parallel n\)                    | 1. Given                                    |
| 2. \(\angle 1 \cong \angle 3\)          | 2. Corresponding Angles Postulate           |
| 3. \(\angle 3 \cong \angle 2\)          | 3. Vertical Angles Theorem                  |
| 4. \(\angle 1 \cong \angle 2\)          | 4. Transitive Property of Congruence        |

### Explanation:

1. The first step is the **given**, stating that lines \(m\) and \(n\) are parallel.
2. By the **Corresponding Angles Postulate**, \(\angle 1\) and \(\angle 3\) are congruent because they are corresponding angles formed by the transversal line intersecting the parallel lines.
3. By the **Vertical Angles Theorem**, \(\angle 3\) and \(\angle 2\) are congruent because they are vertical angles.
4. By the **Transitive Property of Congruence**, \(\angle 1 \cong \angle 3\) and \(\angle 3 \cong \angle 2\) implies \(\angle 1 \cong \angle 2\).

This proves the **Alternate Exterior Angles Theorem**.

Do you have any questions, or would you like more details?

### Related Questions:
1. What is the Corresponding Angles Postulate?
2. How does the Transitive Property of Congruence work?
3. Can alternate interior angles also be congruent when lines are parallel?
4. Why are vertical angles always congruent?
5. How can the Alternate Exterior Angles Theorem be applied to non-parallel lines?

### Tip:
When working with parallel lines and a transversal, always remember to check for congruent angles using postulates like the Corresponding Angles Postulate and the Alternate Interior/Exterior Angles Theorem!

Write a two-column proof of the Alternate Exterior Angles Theorem. Given: m || n, Prove: ∠1 ≅ ∠2

Explore the two-column proof for the Alternate Exterior Angles Theorem, demonstrating that angles formed by a transversal intersecting parallel lines are congruent. This proof involves the Corresponding Angles Postulate, Vertical Angles Theorem, and the Transitive Property of Congruence. Suitable for students in Grades 9-10.

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