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The image shows a two-column proof that needs to be completed, and some steps are already given. The goal is to prove that the sum of two angles, \( \angle 3 \) and \( \angle 4 \), is 90 degrees. Let's analyze the problem and help complete the missing parts.

### Problem Statement:
We are given:
- \( m \angle 2 = 30^\circ \)
- \( m \angle 1 = 2 \times m \angle 2 \)

We need to **prove**:
- \( m \angle 3 + m \angle 4 = 90^\circ \)

### Two-column proof:

| **Statements**                   | **Reasons**            |
|----------------------------------|------------------------|
| 1) \( m \angle 2 = 30 \)         | 1) Given               |
| 2) \( m \angle 1 = 2 \times m \angle 2 \) | 2) Given        |
| 3) \( m \angle 1 = 2 \times 30 = 60 \)  | 3) Substitution      |
| 4) \( m \angle 1 + m \angle 2 = 90 \)   | 4) Linear pair of angles |
| 5) \( m \angle 3 + m \angle 4 = 90 \)   | 5) Vertical angles   |

### Explanation:
1. **Given** \( m \angle 2 = 30^\circ \).
2. **Given** that \( m \angle 1 \) is twice \( m \angle 2 \).
3. Substitute \( m \angle 2 = 30^\circ \) into the second equation, resulting in \( m \angle 1 = 60^\circ \).
4. From the diagram, \( \angle 1 \) and \( \angle 2 \) form a linear pair, meaning their sum is \( 180^\circ \), and since \( m \angle 1 + m \angle 2 = 90^\circ \), the sum of \( m \angle 3 \) and \( m \angle 4 \) must also be 90° as they are vertical angles (equal in measure).
5. Hence, we conclude \( m \angle 3 + m \angle 4 = 90^\circ \).

Let me know if you'd like further details or clarifications!

### Related questions:
1. What is the definition of vertical angles?
2. How do we identify a linear pair of angles in a diagram?
3. Can vertical angles ever be unequal in measure? Why or why not?
4. What is the significance of substitution in a geometric proof?
5. How do we apply the linear pair postulate in other geometry problems?

### Tip:
Always remember that vertical angles are congruent, meaning they have the same measure, which can be useful in many geometry proofs!

Write a two-column proof for the following. Given: m∠2 = 30, m∠1 = 2m∠2. Prove: m∠3 + m∠4 = 90.

Explore how to prove that the sum of two vertical angles, ∠3 and ∠4, equals 90 degrees using the given conditions: m∠2 = 30 and m∠1 = 2m∠2. Learn about linear pairs and vertical angles with step-by-step reasoning for students in Grades 8-10.

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