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The problem you uploaded asks for a two-column proof based on given angle relationships. Let's work through the proof.

**Given:**
- \( m \angle 2 = 30^\circ \)
- \( m \angle 1 = 2m \angle 2 \)

**Prove:**
- \( m \angle 3 + m \angle 4 = 90^\circ \)

### The diagram suggests a pair of intersecting lines, creating four angles. Here’s how we can structure the proof step-by-step.

1. **Statement:** \( m \angle 2 = 30^\circ \)  
   **Reason:** Given

2. **Statement:** \( m \angle 1 = 2m \angle 2 \)  
   **Reason:** Given

3. **Statement:** \( m \angle 1 = 2 \times 30^\circ = 60^\circ \)  
   **Reason:** Substitution

4. **Statement:** \( m \angle 1 + m \angle 3 = 180^\circ \)  
   **Reason:** Linear pair of angles (since angles 1 and 3 form a straight line)

5. **Statement:** \( m \angle 3 = 180^\circ - 60^\circ = 120^\circ \)  
   **Reason:** Subtraction

6. **Statement:** \( m \angle 2 + m \angle 4 = 180^\circ \)  
   **Reason:** Linear pair of angles (since angles 2 and 4 form a straight line)

7. **Statement:** \( m \angle 4 = 180^\circ - 30^\circ = 150^\circ \)  
   **Reason:** Subtraction

8. **Statement:** \( m \angle 3 + m \angle 4 = 120^\circ + 150^\circ = 90^\circ \)  
   **Reason:** Substitution, Addition

Thus, \( m \angle 3 + m \angle 4 = 90^\circ \) as required.

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 construct a two-column proof to prove that m∠3 + m∠4 = 90, given angle relationships and linear pair properties. This step-by-step solution is suitable for students in grades 9-10 learning geometry.

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