https://file.aiquickdraw.com/mathbot/file/1725734355162e8wooeg9.jpg

This is a geometric proof problem that involves the concept of segment congruence. Based on the image, the given information and the figure are:

**Given**:
1. \( GJ \cong JL \)
2. \( GH \cong KL \)

**To Prove**:
\( HI \cong JK \)

We are to construct a two-column proof with "Statements" and "Reasons."

### Proof:

**Statements** | **Reasons**
---|---
1. \( GJ \cong JL \) | 1. Given
2. \( GH \cong KL \) | 2. Given
3. \( HJ = GJ - GH \) | 3. Segment Subtraction (HJ is part of GJ)
4. \( JK = JL - KL \) | 4. Segment Subtraction (JK is part of JL)
5. \( GJ = JL \) | 5. Substitution (from Step 1)
6. \( GH = KL \) | 6. Substitution (from Step 2)
7. \( HJ = JK \) | 7. Substitution and Equality (since the same amounts were subtracted, the resulting segments must be equal)

Thus, we have proven that \( HI \cong JK \).

---

Would you like any details on the steps of this proof or how specific geometric rules apply? Here are some follow-up questions related to this:

1. What is the definition of segment subtraction?
2. Why is congruence used to show equality of segment lengths?
3. Can you explain the properties of midpoint or bisected segments?
4. How do substitution and transitive properties work in proofs?
5. What is the importance of breaking segments into smaller parts?

**Tip**: In geometric proofs, ensure that every statement has a clear reason tied to theorems or definitions.

Given GJ ≅ JL, GH ≅ KL, prove HI ≅ JK using a geometric proof involving segment subtraction.

This problem involves proving segment congruence using geometric properties like segment subtraction. Learn how to prove HI ≅ JK using given congruent segments GJ ≅ JL and GH ≅ KL, suitable for students in Grades 9-11.

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