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To solve for the measures of angles 1 through 7 in the given diagram, let's apply the rules for parallel lines cut by a transversal.

- Lines **m** and **n** are parallel.
- Line **t** is the transversal.

### Angle relationships:
1. **Corresponding angles**: Angles in matching corners when two lines are crossed by another line (transversal). For example, angle 1 and angle 5 are corresponding.
2. **Alternate interior angles**: Angles on opposite sides of the transversal but inside the parallel lines. For example, angle 3 and angle 5 are alternate interior angles.
3. **Vertical angles**: Opposite angles formed by two intersecting lines, and they are always equal. For example, angle 1 and angle 3 are vertical angles.
4. **Supplementary angles**: Angles that add up to 180° because they form a straight line.

### Given:
- **m∠3 = 57°**, as shown in the diagram.

Now let's find the measures of all the angles:

- **m∠1 = m∠3 = 57°** (Vertical angles)
- **m∠5 = m∠1 = 57°** (Corresponding angles)
- **m∠7 = m∠5 = 57°** (Vertical angles)
- **m∠2 = 180° - m∠1 = 180° - 57° = 123°** (Supplementary to angle 1)
- **m∠4 = m∠2 = 123°** (Vertical angles)
- **m∠6 = m∠2 = 123°** (Corresponding angles)

### Final answers:
- m∠1 = 57°
- m∠2 = 123°
- m∠3 = 57°
- m∠4 = 123°
- m∠5 = 57°
- m∠6 = 123°
- m∠7 = 57°

Let me know if you'd like further clarifications or have any questions!

### Relative questions:
1. What are the conditions for lines to be considered parallel?
2. How can you determine if two angles are corresponding in a transversal?
3. What is the relationship between alternate interior angles in a transversal situation?
4. How can vertical angles be identified and calculated?
5. What types of angles are supplementary in parallel lines cut by a transversal?

### Tip:
Always look for vertical, corresponding, and alternate interior angles when working with parallel lines and a transversal!

Find the measure of angles 1-7 given that lines m and n are parallel and t is transversal.

Learn how to find the measures of angles formed by a transversal crossing two parallel lines. This problem involves identifying corresponding, alternate interior, vertical, and supplementary angles. Suitable for students in Grades 7-9.

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