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The image shows a worksheet with problems on **proving segment relationships** in geometry, specifically focusing on the properties of equality and congruence.

Here’s how you can solve and complete the proofs:

### 1. Justify Each Statement:
1. **QA = QA**:
   - **Reason**: Property of Equality (reflexive property: any segment is equal to itself).

2. **If AB ≅ BC and BC ≅ CE, then AB ≅ CE**:
   - **Reason**: Transitive Property of Congruence (if two segments are each congruent to a third, they are congruent to each other).

3. **If Q is between P and R, then PR = PQ + QR**:
   - **Reason**: Segment Addition Postulate (if a point is between two others, the sum of the smaller segments equals the larger segment).

4. **If AB + BC = EF + FG and AB + BC = AC, then EF + FG = AC**:
   - **Reason**: Transitive Property of Equality (if two sums are equal to the same value, they are equal to each other).

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### 2. Complete Each Proof:
**Proof 5** (Given: SU ≅ LR, TU ≅ LN):
1. **Statement (a)**: SU ≅ LR, TU ≅ LN
   - **Reason (a)**: Given
2. **Statement (b)**: SU + TU = ST and LR + LN = NR
   - **Reason (b)**: Definition of Congruence of Segments (when segments are congruent, they have the same length).
3. **Statement (c)**: SU + TU = LR + LN
   - **Reason (c)**: Addition Property of Equality (adding congruent segments results in congruent sums).
4. **Statement (d)**: ST = NR
   - **Reason (d)**: Substitution Property (replacing equal values in the equation).

**Proof 6** (Given: AB ≅ CD):
1. **Statement (a)**: AB ≅ CD
   - **Reason (a)**: Given
2. **Statement (b)**: CD ≅ AB
   - **Reason (b)**: Symmetric Property of Congruence (if one segment is congruent to another, the reverse is also true).
3. **Statement (c)**: Completed
   - **Reason (c)**: Not needed in this proof, but could note reflexivity or definition of congruence if necessary.

Would you like more clarification or further explanation on any of the steps?

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Here are 5 related questions to practice:

1. What is the difference between the transitive property of congruence and the transitive property of equality?
2. How does the segment addition postulate help in geometric proofs?
3. Can two congruent segments be equal in length but not have the same location? Why?
4. What is the role of substitution in geometric proofs?
5. How would you prove that two angles are congruent using properties of congruence?

**Tip**: When working on geometric proofs, always define your given information clearly and build step-by-step using logical properties like substitution, transitivity, and congruence definitions.

Proving Segment Relationships, completing proofs involving properties of equality, congruence, and the segment addition postulate.

This page provides step-by-step solutions for proving segment relationships using properties of equality, congruence, and the segment addition postulate. Key concepts include the Transitive Property and the Segment Addition Postulate, suitable for students in Grades 8-10.

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