https://file.aiquickdraw.com/mathbot/file/1726528672761wciln5y6.jpg

To assist with your proof, let's break it down step by step for the two-column proof format you're working on. 

### Given:
- \( \angle NPQ \cong \angle RPS \)

### To Prove:
- \( \angle NPR \cong \angle QPS \)

### Diagram Analysis:
- In the diagram, points \( N, Q, P, R, \) and \( S \) form different angles with vertex \( P \) at the center. 
- The goal is to show that the angles \( \angle NPR \) and \( \angle QPS \) are congruent using logical steps.

### Two-Column Proof:

| **Statements**                      | **Reasons**                             |
|--------------------------------------|-----------------------------------------|
| 1. \( \angle NPQ \cong \angle RPS \) | 1. Given                                |
| 2. \( \overrightarrow{PQ} \parallel \overrightarrow{PS} \) | 2. Definition of congruent angles (corresponding angles theorem) |
| 3. \( \angle NPR \cong \angle QPS \) | 3. Angle Addition Postulate (subtraction of congruent angles) |

### Explanation:
1. **Statement 1:** You start with the given information, which states that \( \angle NPQ \) and \( \angle RPS \) are congruent.
   
2. **Statement 2:** Since \( \angle NPQ \) and \( \angle RPS \) are corresponding angles formed by parallel lines \( \overrightarrow{PQ} \parallel \overrightarrow{PS} \), they must be congruent by the Corresponding Angles Postulate.

3. **Statement 3:** Finally, by the Angle Addition Postulate, you can subtract equal parts of these angles (the common angle \( \angle P \)) from both sides, leaving you with \( \angle NPR \cong \angle QPS \).

Would you like more details or clarification on any part of this proof? Here are 5 related questions that can expand your understanding:

1. What is the Corresponding Angles Postulate and how does it apply here?
2. How does the Angle Addition Postulate help in proving angles congruent?
3. Why are parallel lines important in this proof?
4. Can you think of other ways to prove angle congruence using different theorems?
5. How can a paragraph proof be structured based on the two-column proof?

**Tip:** Always double-check which theorems and postulates are applicable in the context of your diagram—visualizing them clearly helps streamline your proof!

Prove that ∠NPR ≅ ∠QPS given ∠NPQ ≅ ∠RPS.

Learn how to prove that ∠NPR ≅ ∠QPS given ∠NPQ ≅ ∠RPS in this geometry proof exercise. The solution applies the Corresponding Angles Theorem and the Angle Addition Postulate. Suitable for students in Grades 9-10 studying geometry.

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