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In this problem, we are asked to prove that \( DF = GH \), given the following conditions:

- \( DE = EH \)
- \( EF = GE \)

### Steps for the proof:

1. **Given Conditions**:  
   We know that:
   - \( DE = EH \) (given)
   - \( EF = GE \) (given)

2. **Break Down of Line Segments**:  
   - Since \( DF \) is composed of the segments \( DE \) and \( EF \), we can express:
     \[
     DF = DE + EF
     \]
   - Similarly, since \( GH \) is composed of the segments \( GE \) and \( EH \), we can express:
     \[
     GH = GE + EH
     \]

3. **Substitute the Given Equalities**:  
   - From the given \( DE = EH \), we can replace \( EH \) with \( DE \) in the expression for \( GH \):
     \[
     GH = GE + DE
     \]
   - From the given \( EF = GE \), we can replace \( GE \) with \( EF \):
     \[
     GH = EF + DE
     \]

4. **Compare the Expressions for \( DF \) and \( GH \)**:  
   - Now we have:
     \[
     DF = DE + EF
     \]
     and
     \[
     GH = EF + DE
     \]
   - Since both expressions for \( DF \) and \( GH \) are identical (just written in reverse order), we conclude:
     \[
     DF = GH
     \]

Thus, we have proven that \( DF = GH \).

---

Do you have any questions or would you like more details on any step?

### Related Questions:
1. How can we prove that two line segments are equal using the transitive property?
2. Can this problem be solved using coordinate geometry?
3. How would the proof change if the diagram was altered (e.g., if the lines weren't straight)?
4. How do the properties of congruent triangles relate to this proof?
5. How does vector addition relate to line segments like \( DF \) and \( GH \)?

**Tip**: When solving geometric proofs, always write down known relationships and try to express unknown quantities in terms of known ones.

Use the given information to prove that DF = GH.

Explore the geometric proof where DF equals GH based on equal segments conditions. This problem involves understanding the segment addition postulate and using it to prove equality of two line segments. Suitable for high school students.

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