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To complete the proof that \( \triangle GIK \cong \triangle GJH \), we will fill in the missing reasons for the statements provided in the table.

### Step-by-Step Reasoning for the Proof:

1. **Statement**: \( \angle JGK \cong \angle HGI \)  
   **Reason**: *Vertical angles are congruent*.  
   - The angles formed by the intersection of two lines are equal, and since \( \angle JGK \) and \( \angle HGI \) are opposite angles formed by the intersection of lines, they are congruent.

2. **Statement**: \( \overline{GI} \cong \overline{GJ} \)  
   **Reason**: *Given*  
   - This is given in the problem as part of the conditions for the proof.

3. **Statement**: \( \angle GIK \cong \angle GJH \)  
   **Reason**: *Alternate interior angles are congruent*  
   - Since \( \overline{IK} \parallel \overline{JH} \) and \( \overline{GI} \) is a transversal, alternate interior angles \( \angle GIK \) and \( \angle GJH \) are congruent.

4. **Statement**: \( m\angle IGK = m\angle IGJ + m\angle JGK \)  
   **Reason**: *Angle addition postulate*  
   - The measure of an angle is the sum of its smaller angles. So, the measure of \( \angle IGK \) is the sum of \( \angle IGJ \) and \( \angle JGK \).

5. **Statement**: \( m\angle HGJ = m\angle HGI + m\angle IGJ \)  
   **Reason**: *Angle addition postulate*  
   - Similarly, the measure of \( \angle HGJ \) is the sum of \( \angle HGI \) and \( \angle IGJ \).

6. **Statement**: \( m\angle IGK = m\angle IGJ + m\angle HGI \)  
   **Reason**: *Substitution*  
   - From step 1, we know \( \angle JGK \cong \angle HGI \), so we substitute \( m\angle JGK \) with \( m\angle HGI \).

7. **Statement**: \( m\angle HGJ = m\angle IGK \)  
   **Reason**: *Transitive property*  
   - By the transitive property of equality, since both \( m\angle IGK \) and \( m\angle HGJ \) equal the same expression, they are congruent.

8. **Statement**: \( \triangle GIK \cong \triangle GJH \)  
   **Reason**: *ASA (Angle-Side-Angle) Congruence*  
   - Two triangles are congruent if two angles and the included side are congruent. Here, we have \( \angle GIK \cong \angle GJH \), \( \overline{GI} \cong \overline{GJ} \), and \( \angle JGK \cong \angle HGI \).

### Completed Table:

| Statement                                      | Reason                           |
|------------------------------------------------|----------------------------------|
| 1. \( \angle JGK \cong \angle HGI \)           | Vertical angles are congruent    |
| 2. \( \overline{GI} \cong \overline{GJ} \)     | Given                            |
| 3. \( \angle GIK \cong \angle GJH \)           | Alternate interior angles        |
| 4. \( m\angle IGK = m\angle IGJ + m\angle JGK \) | Angle addition postulate         |
| 5. \( m\angle HGJ = m\angle HGI + m\angle IGJ \) | Angle addition postulate         |
| 6. \( m\angle IGK = m\angle IGJ + m\angle HGI \) | Substitution                     |
| 7. \( m\angle HGJ = m\angle IGK \)             | Transitive property              |
| 8. \( \triangle GIK \cong \triangle GJH \)     | ASA Congruence                   |

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Do you need further details or have any questions about the proof?

### Related Questions:
1. How does the Angle-Side-Angle (ASA) congruence postulate work?
2. Can you explain how vertical angles are formed and why they are congruent?
3. What is the transitive property, and how is it used in geometry proofs?
4. What is the significance of alternate interior angles in parallel lines?
5. How does the angle addition postulate help in geometry proofs?

**Tip**: Understanding the congruence postulates (ASA, SAS, SSS, and AAS) is key to solving triangle proofs.

Complete the proof that triangle GIK is congruent to triangle GJH, using the provided statements and reasons.

Complete the proof that triangle GIK is congruent to triangle GJH using geometric concepts such as vertical angles, alternate interior angles, and the Angle-Side-Angle (ASA) postulate. This proof is suitable for students in grades 9-12.

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