This is a two-column proof requiring 6 answers for each column.

Given: line AB is perpendicular to line CD.
Prove: m angle 2 + m angle 4 + m angle 7 = m angle 1.

We have no idea what those angles are from your data.

To prove: m∠2 + m∠4 + m∠7 = m∠1.

In order to prove this statement, we need to use the given information that line AB is perpendicular to line CD.

Let's break down the steps to prove this statement using a two-column proof format with six answers for each column:

| Statements | Reasons |
|------------------------|-------------------------------------------|
| 1. AB is perpendicular to CD. | Given |
| 2. m∠2 + m∠3 = 90°. | Definition of perpendicular lines |
| 3. m∠3 + m∠4 + m∠7 = 180°. | Angle addition postulate (sum of angles on a straight line) |
| 4. m∠5 + m∠6 = 90°. | Definition of perpendicular lines |
| 5. m∠5 + m∠6 + m∠7 = 180°. | Angle addition postulate |
| 6. m∠2 + m∠3 + m∠4 + m∠5 + m∠6 + m∠7 = 360°. | Exterior angle theorem (sum of exterior angles of a polygon) |
| 7. m∠2 + m∠4 + m∠7 = m∠1. | Substitution (using angles 3, 5, and 6) |

Let's go through the reasoning behind each statement:

1. AB is perpendicular to CD. (Given)
This is given information, which states that line AB forms a 90° angle with line CD.

2. m∠2 + m∠3 = 90°. (Definition of perpendicular lines)
By the definition of perpendicular lines, the sum of the measures of the angles formed by two perpendicular lines is always 90°.

3. m∠3 + m∠4 + m∠7 = 180°. (Angle addition postulate)
This is the angle addition postulate, which states that the sum of the measures of the angles formed by collinear rays that share a common endpoint is 180°.

4. m∠5 + m∠6 = 90°. (Definition of perpendicular lines)
Similar to statement 2, this follows from the definition of perpendicular lines.

5. m∠5 + m∠6 + m∠7 = 180°. (Angle addition postulate)
Again, using the angle addition postulate, we see that the sum of the measures of angles formed by collinear rays that share a common endpoint is 180°.

6. m∠2 + m∠3 + m∠4 + m∠5 + m∠6 + m∠7 = 360°. (Exterior angle theorem)
The exterior angle theorem states that the sum of the measures of the exterior angles of any polygon is always 360°.

7. m∠2 + m∠4 + m∠7 = m∠1. (Substitution)
Using the previously established relationships between the angles (statements 2, 3, 4, 5, and 6), we substitute the values to obtain the desired equation.

Therefore, we have proven that m∠2 + m∠4 + m∠7 = m∠1 using a two-column proof.