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To solve this problem, let's analyze the given diagram and relationships between the angles:

We know:
- Two parallel lines are cut by a transversal.
- The measure of angle \( m \angle 1 = 57^\circ \).

### Step-by-Step Analysis:

1. **Alternate Interior Angles Theorem:**
   Since the lines are parallel and the transversal cuts through them, we can apply the alternate interior angles theorem. According to this theorem:
   - \( \angle 1 \) and \( \angle 6 \) are alternate interior angles, so \( m \angle 6 = m \angle 1 = 57^\circ \).

2. **Corresponding Angles Theorem:**
   According to this theorem:
   - \( \angle 1 \) and \( \angle 7 \) are corresponding angles, so \( m \angle 7 = m \angle 1 = 57^\circ \).

### Final Answer:
- \( m \angle 6 = 57^\circ \)
- \( m \angle 7 = 57^\circ \)

Would you like a detailed explanation of how these theorems work? 

Here are some related questions:
1. What are alternate exterior angles in the same type of diagram?
2. How would the solution change if \( m \angle 1 \) were a different angle?
3. What other angles can we determine from the given information in this diagram?
4. How do vertical angles relate to parallel lines cut by a transversal?
5. Can you identify any supplementary angles in this diagram?

**Tip:** Always remember that when two parallel lines are cut by a transversal, alternate interior angles and corresponding angles are equal!

Suppose m∠1 = 57°. Find m∠6 and m∠7.

In this geometry problem, learn how to find the measures of angles 6 and 7 when two parallel lines are cut by a transversal. Theorems like alternate interior angles and corresponding angles are used. Given m∠1 = 57°, we can conclude that m∠6 = 57° and m∠7 = 57°. Suitable for students in Grades 8-10.

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