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To complete the two-column proof, let's analyze the given information and determine the correct reasoning to prove that \( KA \cong JY \).

### Given Information:
1. \( RA \cong NY \)
2. \( \angle R \cong \angle N \)
3. \( \angle A \cong \angle Y \)

We need to prove that \( KA \cong JY \).

### Proof Analysis:

1. **Statement 1**: \( RA \cong NY \), \( \angle R \cong \angle N \), \( \angle A \cong \angle Y \)
   - **Reason**: Given

2. **Statement 2**: \( \triangle KRA \cong \triangle JNY \)
   - **Reason**: By **ASA (Angle-Side-Angle) Congruence Postulate**. We have two pairs of congruent angles (\( \angle R \cong \angle N \), \( \angle A \cong \angle Y \)) and the included side \( RA \cong NY \), so the triangles are congruent.

3. **Statement 3**: \( KA \cong JY \)
   - **Reason**: By **CPCTC (Corresponding Parts of Congruent Triangles are Congruent)**. Since \( \triangle KRA \cong \triangle JNY \), all corresponding parts of these triangles are congruent, including \( KA \) and \( JY \).

### Completed Proof:

| Statements                                    | Reason                                      |
|-----------------------------------------------|---------------------------------------------|
| 1. \( RA \cong NY \), \( \angle R \cong \angle N \), \( \angle A \cong \angle Y \) | 1. Given                                     |
| 2. \( \triangle KRA \cong \triangle JNY \)    | 2. ASA Congruence Postulate                 |
| 3. \( KA \cong JY \)                          | 3. CPCTC                                     |

This completes the proof that \( KA \cong JY \).

Would you like more details on any step or have any questions?

Here are five related questions to deepen your understanding:
1. What are the different congruence postulates that can prove triangle congruence?
2. How does the ASA postulate differ from the SAS postulate?
3. Why is CPCTC a valid reason in triangle congruence proofs?
4. Could the proof change if one of the angles was not congruent?
5. What other methods can be used to prove two segments congruent in geometry?

**Tip**: When proving triangles congruent, focus on identifying pairs of congruent angles and sides to decide which postulate (like ASA, SSS, SAS, or AAS) applies.

Complete the two-column proof by filling in the blanks for the given congruence relationships between triangles.

Learn how to complete a two-column proof by using the ASA (Angle-Side-Angle) Congruence Postulate and CPCTC (Corresponding Parts of Congruent Triangles are Congruent) to prove congruence between segments in given triangles. Suitable for students in Grades 9-10.

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