https://file.aiquickdraw.com/mathbot/file/17303919600931k9flkfl.jpg

To complete the proof that \( \triangle HIK \cong \triangle JGK \), let's examine each statement and reason carefully:

### Given Statements:
1. \( K \) is the midpoint of \( \overline{GI} \) - **Reason: Given**
2. \( K \) is the midpoint of \( \overline{HJ} \) - **Reason: Given**
3. \( \overline{GJ} \cong \overline{HI} \) - **Reason: Given**

### Completing the Proof:
4. \( \overline{GK} \cong \overline{IK} \) - **Reason: Definition of midpoint**
   - Since \( K \) is the midpoint of \( \overline{GI} \), segments \( \overline{GK} \) and \( \overline{IK} \) are equal.

5. \( \overline{HK} \cong \overline{JK} \) - **Reason: Definition of midpoint**
   - Similarly, since \( K \) is the midpoint of \( \overline{HJ} \), segments \( \overline{HK} \) and \( \overline{JK} \) are equal.

6. \( \triangle HIK \cong \triangle JGK \) - **Reason: Side-Side-Side (SSS) Congruence Postulate**
   - Since all three pairs of corresponding sides are congruent (\( \overline{HI} \cong \overline{GJ} \), \( \overline{IK} \cong \overline{GK} \), and \( \overline{HK} \cong \overline{JK} \)), the triangles are congruent by the SSS postulate.

This completes the proof that \( \triangle HIK \cong \triangle JGK \) by the SSS Congruence Postulate.

Would you like further details on any of these steps?

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Here are five related questions to expand understanding:
1. What is the SSS Congruence Postulate, and why does it work for proving triangles congruent?
2. How would the proof change if \( K \) was not a midpoint but another point on \( \overline{GI} \) and \( \overline{HJ} \)?
3. Can this type of proof be applied to other geometric figures, such as quadrilaterals?
4. What other congruence postulates or theorems could be used if we didn’t have SSS?
5. How can we use congruent triangles to solve real-world geometry problems?

**Tip:** Always identify corresponding parts of the triangles carefully, as misidentification can lead to incorrect conclusions in proofs.

K is the midpoint of GI and HJ. Complete the proof that ΔHIK ≅ ΔJGK.

Explore a geometry problem where K is the midpoint of GI and HJ, and complete the proof that ΔHIK is congruent to ΔJGK. This solution uses the Side-Side-Side (SSS) Congruence Postulate, suitable for students in Grades 8-10.

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