Geometry proof problem with given information and steps to prove parallel lines.

You're working on a two-column proof to show that lines m and n are parallel, given that ∠1 ≅ ∠2.

Let’s walk through the logic step-by-step and fill in the missing statements and reasons.

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🧩 Given:

- ∠1 ≅ ∠2

🎯 Goal:

- Prove: m ∥ n

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## ✔ Step-by-Step Proof:

We’ll use the Vertical Angles Theorem and the Transitive Property of Congruence, then apply the Corresponding Angles Postulate (or its converse) to conclude the lines are parallel.

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🔢 Filled-in Two-Column Proof:

| Statements | Reasons |
|--------------------------|----------------------------------|
| 1. ∠1 ≅ ∠2 | Given |
| 2. ∠2 ≅ ∠3 | Vertical Angles Theorem |
| 3. ∠1 ≅ ∠3 | Transitive Property of Congruence|
| 4. m ∥ n | Corresponding Angles Converse |

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## 📚 Explanation:

Step 1: ∠1 ≅ ∠2 — *Given*

This is stated in the problem. We start here.

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Step 2: ∠2 ≅ ∠3 — *Vertical Angles Theorem*

Look at the diagram: ∠2 and ∠3 are vertical angles (they are opposite each other when two lines intersect).
👉 Vertical angles are always congruent.

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Step 3: ∠1 ≅ ∠3 — *Transitive Property of Congruence*

If ∠1 ≅ ∠2 and ∠2 ≅ ∠3, then by the transitive property, ∠1 ≅ ∠3.

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Step 4: m ∥ n — *Corresponding Angles Converse*

Now look at ∠1 and ∠3 — they are corresponding angles formed by the transversal crossing lines m and n.

> 📌 Corresponding Angles Converse Postulate: If two lines are cut by a transversal so that corresponding angles are congruent, then the lines are parallel.

Since ∠1 ≅ ∠3 (corresponding angles), we can conclude m ∥ n.

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## ✔ Final Answer:

The completed proof is:

| Statements | Reasons |
|-------------------|----------------------------------|
| ∠1 ≅ ∠2 | Given |
| ∠2 ≅ ∠3 | Vertical Angles Theorem |
| ∠1 ≅ ∠3 | Transitive Property of Congruence|
| m ∥ n | Corresponding Angles Converse |

✔ This proves that lines m and n are parallel.

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Let me know if you’d like to see a diagram explanation or how this connects to other angle relationships!