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To solve this proof, we are given that \( \angle LKM \cong \angle JKM \) and need to prove that \( \triangle LKM \cong \triangle JKM \).

Here's a breakdown of the two-column proof with the missing reasons:

1. **Statement:** \( \angle LKM \cong \angle JKM \)  
   **Reason:** Given.

2. **Statement:** \( KM \cong KM \)  
   **Reason:** Reflexive Property (since \( KM \) is the same segment for both triangles).

3. **Statement:** \( \triangle LKM \cong \triangle JKM \)  
   **Reason:** Angle-Side-Angle (ASA) Congruence Postulate (because we have two pairs of congruent angles and the included side \( KM \)).

### Completed Two-Column Proof

| Statements                 | Reasons                    |
|----------------------------|----------------------------|
| 1. \( \angle LKM \cong \angle JKM \) | Given                      |
| 2. \( KM \cong KM \)                | Reflexive Property         |
| 3. \( \triangle LKM \cong \triangle JKM \) | ASA Congruence Postulate |

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

### Related Questions

1. Why does the ASA Congruence Postulate apply here?
2. What are the alternative ways to prove triangle congruence?
3. How does the Reflexive Property help in proving congruence?
4. What other properties might be useful in congruent triangle proofs?
5. How do we determine which congruence postulate to use?

### Tip
When proving triangles congruent, focus on identifying pairs of congruent angles and sides that match one of the postulates (SSS, SAS, ASA, AAS). This will simplify the proof process.

Complete the two-column proof by filling in the blanks.

Learn how to complete a two-column proof for proving triangle congruence using the ASA Congruence Postulate and the Reflexive Property. This problem involves identifying congruent angles and a shared side. Suitable for students in Grades 8-10.

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