https://file.aiquickdraw.com/mathbot/file/17293521779632mqr20tz.jpg

Let's work through this two-column proof and fill in the blanks step by step:

### Given Information:
- \( \angle S \cong \angle P \)
- \( \angle WSJ \) is a right angle

### Prove: 
- \( \triangle SJW \cong \triangle PJW \)

#### Step-by-step:

1. **Statement: \( \angle S \cong \angle P \)**  
   **Reason:** Given

2. **Statement: \( \angle WSJ \) is a right angle**  
   **Reason:** Given

3. **Statement:** \( \angle WJP \cong \angle WJS \)  
   **Reason:** The supplement is a right angle, and all right angles are congruent.

4. **Statement:** \( \overline{WJ} \cong \overline{WJ} \)  
   **Reason:** Reflexive Property (a segment is congruent to itself).

5. **Statement:** \( \triangle SJW \cong \triangle PJW \)  
   **Reason:** **Angle-Angle-Side (AAS)** Congruence Theorem (two angles and the non-included side are congruent).

Here’s the complete proof:

| Statements                              | Reasons                                               |
|-----------------------------------------|-------------------------------------------------------|
| \( \angle S \cong \angle P \)           | 1. Given                                              |
| \( \angle WSJ \) is a right angle       | 2. Given                                              |
| \( \angle WJP \cong \angle WJS \)       | 3. The supplement is a right angle and right angles are congruent. |
| \( \overline{WJ} \cong \overline{WJ} \) | 4. Reflexive Property                                 |
| \( \triangle SJW \cong \triangle PJW \) | 5. AAS (Angle-Angle-Side) Congruence Theorem          |

Would you like further details or have any questions about the solution?

Here are 5 relative questions:
1. How does the Angle-Angle-Side (AAS) theorem work in proving congruent triangles?
2. What other congruence theorems exist aside from AAS?
3. Why is the Reflexive Property important in geometry proofs?
4. How do we identify corresponding parts in congruent triangles?
5. What does it mean for angles to be supplements of each other?

**Tip:** When dealing with congruent triangles, always look for congruent angles and sides first to determine the best congruence theorem to apply.

Complete the two-column proof by filling in the blanks:
Given: ∠S ≅ ∠P, ∠WSJ is a right angle
Prove: ΔSJW ≅ ΔPJW

Learn how to complete a two-column proof to demonstrate triangle congruence using the Angle-Angle-Side (AAS) Congruence Theorem. This step-by-step example involves congruent angles, the reflexive property, and right angle congruence. Suitable for students in Grades 8-10.

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