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To complete the proof that \( \triangle GJK \cong \triangle IHK \), we need to fill in the missing reasons for each statement in the proof. Here’s the completed proof:

### Proof Structure
| Statement | Reason |
|-----------|--------|
| **1.** \( K \) is the midpoint of \( \overline{GI} \) | Given |
| **2.** \( \overline{GJ} \parallel \overline{HI} \) | Given |
| **3.** \( \angle G \cong \angle I \) | Alternate Interior Angles Theorem (since \( \overline{GJ} \parallel \overline{HI} \)) |
| **4.** \( \angle H \cong \angle J \) | Alternate Interior Angles Theorem (same reasoning as above) |
| **5.** \( \overline{GK} \cong \overline{IK} \) | Definition of Midpoint (since \( K \) is the midpoint of \( \overline{GI} \)) |
| **6.** \( \triangle GJK \cong \triangle IHK \) | ASA (Angle-Side-Angle) Congruence Postulate |

### Explanation of the Congruence
We used the **ASA (Angle-Side-Angle) Congruence Postulate** because:
1. We have two pairs of congruent angles: \( \angle G \cong \angle I \) and \( \angle H \cong \angle J \) by the Alternate Interior Angles Theorem.
2. The side \( \overline{GK} \cong \overline{IK} \) is congruent by the definition of a midpoint.

Would you like further details on any step or another explanation?

### Follow-up Questions
1. What is the significance of the Alternate Interior Angles Theorem in this proof?
2. How does the midpoint property contribute to the congruence of triangles?
3. Why is the ASA postulate applicable in proving triangle congruence?
4. Can this proof be completed with other congruence postulates, like SAS or SSS? Why or why not?
5. How can we verify that two triangles are congruent by observing their parallel sides?

### Tip:
When proving triangles congruent, look for given parallel lines or midpoints—they often provide angle congruence or segment congruence essential for postulates like ASA, SAS, or SSS.

K is the midpoint of GI. Complete the proof that ΔGJK ≅ ΔIHK.

Complete the proof that triangles ΔGJK and ΔIHK are congruent using the ASA postulate. This problem involves concepts of midpoints, parallel lines, and the Alternate Interior Angles Theorem. Suitable for high school geometry students in grades 9-10.

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