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The problem you provided is a geometric proof that aims to show that segments \( \overline{CE} \cong \overline{EG} \). The given information includes:

1. \( CD = EF \)
2. \( DE = FG \)

The proof involves using the Segment Addition Postulate. Here’s how the rest of the proof can be completed:

### Proof

| Statements                     | Reasons                       |
|---------------------------------|-------------------------------|
| 1. \( CD = EF \)                | Given                         |
| 2. \( DE = FG \)                | Given                         |
| 3. \( CD + DE = EF + FG \)      | Segment Addition Postulate     |
| 4. \( CE = EG \)                | Substitution (since \( CE = CD + DE \) and \( EG = EF + FG \)) |
| 5. \( \overline{CE} \cong \overline{EG} \) | Definition of Congruence of Segments |

This proof shows that the two segments \( \overline{CE} \) and \( \overline{EG} \) are congruent by applying the Segment Addition Postulate and then recognizing that \( CE \) and \( EG \) are equal in length, hence the two segments are congruent.

### Do you have any questions or want further details?

Here are five relative questions that build on this problem:

1. What is the Segment Addition Postulate, and why is it useful in geometric proofs?
2. How would the proof change if one of the given equalities, such as \( CD = EF \), did not hold?
3. Can you think of a real-world example where the Segment Addition Postulate might be applied?
4. How does the concept of congruence differ from the concept of equality in geometry?
5. What other theorems or postulates could help prove segment congruence?

**Tip:** In geometry, always make sure to distinguish between "equal" (referring to numerical values) and "congruent" (referring to shapes or segments that are identical in size but may have different positions).

Given: CD = EF, DE = FG. Prove: CE ≅ EG.

Learn how to prove that segments CE and EG are congruent using the Segment Addition Postulate. This geometric proof involves given equalities CD = EF and DE = FG, and is suitable for students in Grades 9-10.

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