https://file.aiquickdraw.com/mathbot/file/1730308846254rzkp6vuu.jpg

To complete this proof, let’s look at the missing reason for statement 5:

### Proof Details:

The goal is to prove that \(\triangle HIK \cong \triangle JGK\).

**Statements and Reasons**:

1. **Statement**: \(K\) is the midpoint of \(\overline{GI}\)  
   **Reason**: Given

2. **Statement**: \(K\) is the midpoint of \(\overline{HJ}\)  
   **Reason**: Given

3. **Statement**: \(\overline{HI} \cong \overline{GJ}\)  
   **Reason**: Given

4. **Statement**: \(\overline{GK} \cong \overline{IK}\)  
   **Reason**: Definition of midpoint (Since \(K\) is the midpoint of \(\overline{GI}\))

5. **Statement**: \(\overline{HK} \cong \overline{JK}\)  
   **Reason**: Definition of midpoint (Since \(K\) is the midpoint of \(\overline{HJ}\))

6. **Statement**: \(\triangle HIK \cong \triangle JGK\)  
   **Reason**: SSS (Side-Side-Side congruence)

### Explanation

Since \(K\) is the midpoint of both \(\overline{GI}\) and \(\overline{HJ}\), it divides these segments into equal parts, which establishes congruency of the pairs of sides \(GK = IK\) and \(HK = JK\). Coupled with the given \(HI = GJ\), we can conclude congruency between \(\triangle HIK\) and \(\triangle JGK\) by the SSS criterion.

Would you like a more detailed explanation of any step?

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### Related Questions

1. What other congruence criteria (e.g., SAS, ASA) could be used to prove triangle congruence?
2. How does the midpoint property help in establishing segment congruence?
3. Can the angle properties of triangles help in this proof?
4. What if \(K\) was not a midpoint? How would the proof change?
5. How is the Side-Side-Side (SSS) congruence postulate applied in different types of proofs?

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### Tip

In geometry proofs, always start by identifying given information, as it often leads directly to what you need to prove.

K is the midpoint of segments GI and HJ. Complete the proof that triangle HIK is congruent to triangle JGK.

Explore a step-by-step proof to show that triangle HIK is congruent to triangle JGK using the SSS Congruence Theorem and midpoint properties. Ideal for students in Grades 8-10, this problem helps reinforce geometry concepts of triangle congruence and midpoints.

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