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The problem is asking for a two-column proof of the **Alternate Exterior Angles Theorem** based on the given diagram where \( m \parallel n \), and we need to prove that \( \angle 1 \cong \angle 2 \).

### The Alternate Exterior Angles Theorem states:
- If two parallel lines are cut by a transversal, then each pair of alternate exterior angles is congruent.

### Here's how the two-column proof can be completed:

#### **Given:**
- \( m \parallel n \)

#### **Prove:**
- \( \angle 1 \cong \angle 2 \)

### Two-column Proof:

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

### Explanation of the Proof:
1. **Given** that the two lines \( m \) and \( n \) are parallel.
2. By the **Alternate Interior Angles Theorem**, \( \angle 1 \) and \( \angle 3 \) are congruent because they lie on opposite sides of the transversal and between the parallel lines.
3. Using the **Vertical Angles Theorem**, \( \angle 3 \) and \( \angle 2 \) are congruent because they are vertical angles formed by the intersection of the transversal and line \( n \).
4. Finally, by the **Transitive Property of Congruence**, since \( \angle 1 \cong \angle 3 \) and \( \angle 3 \cong \angle 2 \), we can conclude that \( \angle 1 \cong \angle 2 \).

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Would you like further clarification or more details on any step?

Here are some related questions for deeper understanding:
1. How are alternate interior angles related to parallel lines?
2. What is the transitive property in geometry, and how is it applied here?
3. Can the Alternate Exterior Angles Theorem be proved using the Corresponding Angles Postulate?
4. What is the difference between alternate interior and alternate exterior angles?
5. How do vertical angles arise, and why are they always congruent?

**Tip:** When working with parallel lines and a transversal, always look for alternate interior and exterior angles first, as they often provide key relationships to prove congruence.

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

Explore the step-by-step two-column proof of the Alternate Exterior Angles Theorem, proving that ∠1 ≅ ∠2 using parallel lines and transversals. Learn how the Alternate Interior Angles Theorem, Vertical Angles Theorem, and Transitive Property are applied in this proof.

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