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To prove that \( \angle 7 \cong \angle 5 \) given that \( r \parallel s \), we can use properties of parallel lines and transversal angles.

Here's the reasoning in steps:

1. **Given**: \( r \parallel s \).

2. Since lines \( r \) and \( s \) are parallel, and the line passing through angles 3, 4, 5, and 6 is a transversal, we can apply the concept of corresponding angles.

3. **Corresponding Angles**: According to the corresponding angles postulate, when a transversal crosses two parallel lines, each pair of corresponding angles is congruent. Therefore, \( \angle 3 \cong \angle 5 \).

4. Similarly, the line passing through angles 1, 2, 7, and 8 is also a transversal to the parallel lines \( r \) and \( s \). Using the same postulate, we find that \( \angle 3 \cong \angle 7 \) because they are also corresponding angles.

5. **Transitive Property**: Since \( \angle 3 \cong \angle 5 \) and \( \angle 3 \cong \angle 7 \), we can conclude by the transitive property that \( \angle 7 \cong \angle 5 \).

### Completed Proof:
- **Statement**: \( r \parallel s \)
  - **Reason**: Given
- **Statement**: \( \angle 3 \cong \angle 5 \)
  - **Reason**: Corresponding angles with parallel lines \( r \parallel s \)
- **Statement**: \( \angle 3 \cong \angle 7 \)
  - **Reason**: Corresponding angles with parallel lines \( r \parallel s \)
- **Statement**: \( \angle 7 \cong \angle 5 \)
  - **Reason**: Transitive property of congruence

Would you like further details or have any questions?

### Related Questions
1. What are corresponding angles, and why are they congruent?
2. How does the transitive property work in angle congruence?
3. How does the alternate interior angles theorem apply with parallel lines?
4. What is the difference between alternate interior angles and corresponding angles?
5. How can we prove that two angles are supplementary?

### Tip:
When working with parallel lines and a transversal, always identify pairs of corresponding, alternate interior, or alternate exterior angles to find congruence or supplementary relationships.

Use the given information to prove that ∠7 ≅ ∠5, given r || s.

Understand how to prove that ∠7 ≅ ∠5 using the corresponding angles postulate and transitive property with parallel lines and a transversal. This problem is designed for students in Grades 8-10 studying geometry.

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