https://file.aiquickdraw.com/mathbot/file/17271811761056sv48k39.jpg

Let's work through the proof step by step based on the diagram provided. Here are the correct logical reasons for each statement:

1. **C is the midpoint of BD and AE.**
   - **Reason:** Definition of midpoint (Given).

2. **BC = CD and AC = CE**
   - **Reason:** Definition of midpoint (Def. of mdpt).

3. **AC = AB + BC and CE = CD + DE**
   - **Reason:** Segment Addition Postulate (Seg. Add. Post.).

4. **AC − BC = AB**
   - **Reason:** Subtraction Property of Equality (Subtr. Prop. (1st use)).

5. **CE − CD = DE**
   - **Reason:** Subtraction Property of Equality (Subtr. Prop. (2nd use)).

6. **CE − CD = AB**
   - **Reason:** Substitution Property (Substitution Property).

7. **AB = CE − CD**
   - **Reason:** Symmetric Property of Equality (Symmetric Property of Equality).

8. **AB = DE**
   - **Reason:** Transitive Property (Trans. Prop.).

The completed proof shows that \( AB = DE \).

Would you like more details on any part of this proof?

Here are five related questions to consider:
1. What is the significance of the midpoint in a line segment?
2. How does the Segment Addition Postulate apply to dividing a line segment?
3. Can you explain how the Subtraction Property of Equality works in geometrical proofs?
4. How does the Transitive Property ensure equality in geometrical relationships?
5. What are other applications of the Symmetric Property of Equality in geometry?

**Tip:** When breaking down proofs, focus on how properties like midpoint definitions and segment addition work together. This helps in structuring logical steps.

Five lights, A, B, C, D, and E, are aligned in a row. The middle light is the midpoint of the segment between the second and fourth lights and also the midpoint of the segment between the first and last lights. Prove AB = DE.

In this geometry problem, five lights A, B, C, D, and E are aligned in a row with specific midpoints. Follow a detailed step-by-step proof to show that AB equals DE using properties of equality, the midpoint definition, and the Segment Addition Postulate. Suitable for students in Grades 9-12.

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