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In this question, you are given that lines \( p \parallel q \), and that \( \angle 1 \cong \angle 5 \). You are asked to prove that \( \angle 2 \cong \angle 5 \), using a proof with statements and reasons.

Let's break down the solution step by step.

1. **Given**: \( p \parallel q \)
   - Reason: Given
2. **Given**: \( \angle 1 \cong \angle 5 \)
   - Reason: Given
3. **\( \angle 2 \cong \angle 1 \)** 
   - Reason: This follows from the *Alternate Interior Angles Theorem*. Since \( p \parallel q \), and angles 1 and 2 are on opposite sides of the transversal, they are congruent.
4. **\( \angle 2 \cong \angle 5 \)** 
   - Reason: This follows from the *Transitive Property*. Since \( \angle 2 \cong \angle 1 \) and \( \angle 1 \cong \angle 5 \), we can conclude that \( \angle 2 \cong \angle 5 \).

The reasoning provided for each step is important to ensure the logical flow of the proof. Here’s the complete proof:

| **Statements**              | **Reasons**                       |
|-----------------------------|------------------------------------|
| \( p \parallel q \)          | Given                             |
| \( \angle 1 \cong \angle 5 \) | Given                             |
| \( \angle 2 \cong \angle 1 \) | Alternate Interior Angles Theorem |
| \( \angle 2 \cong \angle 5 \) | Transitive Property               |

Would you like further clarification on any of the steps, or do you want more details on how these angle theorems work?

Here are five related questions:
1. What is the Alternate Interior Angles Theorem, and how is it applied in different scenarios?
2. Can the Transitive Property be used in non-congruent angle cases? How?
3. How does the relationship between parallel lines and transversals affect angle congruency?
4. What are other angle theorems that can be used in proofs involving parallel lines?
5. How would the proof change if the lines were not parallel?

**Tip**: When working with geometry proofs, always ensure that you identify the relationships between angles using known theorems or properties such as congruence or parallel line theorems.

Given: p || q, angle 1 ≅ angle 5. Prove: angle 2 ≅ angle 5.

This problem involves proving that angle 2 is congruent to angle 5, given parallel lines p and q and the fact that angle 1 is congruent to angle 5. The solution applies the Alternate Interior Angles Theorem and the Transitive Property of Congruence. Suitable for students in Grades 8-10.

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