https://file.aiquickdraw.com/mathbot/file/1725932466828twyscm3c.jpg

Let's break down and complete the two-column proof based on the image provided. The goal is to prove that \( \overline{AC} \cong \overline{CD} \), given that \( C \) is the midpoint of both \( \overline{AE} \) and \( \overline{BD} \), and \( \overline{AE} = \overline{BD} \).

### Proof Outline
We will complete the proof using the steps provided, matching the statements to their corresponding reasons:

1. **Statement: \( C \) is the midpoint of \( \overline{AE} \), \( C \) is the midpoint of \( \overline{BD} \).**
   - **Reason: "Given"**

2. **Statement: \( AC = CE \), \( BC = CD \).**
   - **Reason: "Definition of midpoint"** (since \( C \) is the midpoint, it divides the segments into two equal parts)

3. **Statement: \( AE = BD \).**
   - **Reason: "Given"**

4. **Statement: \( AC + CE = CD + BD \).**
   - **Reason: "Segment Addition Property"** (this property allows us to add segment parts)

5. **Statement: \( AC + CE = BC + CD \).**
   - **Reason: "Substitution Property (1st use)"** (substitute from statement 2 into statement 4)

6. **Statement: \( AC + AC = CD + CD \).**
   - **Reason: "Substitution Property (2nd use)"** (since \( AC = CE \) and \( BC = CD \), substitute these values)

7. **Statement: \( 2AC = 2CD \).**
   - **Reason: "Simplify"** (combine like terms on both sides of the equation)

8. **Statement: \( AC = CD \).**
   - **Reason: "Division Property"** (divide both sides of the equation by 2)

9. **Statement: \( \overline{AC} \cong \overline{CD} \).**
   - **Reason: "Definition of congruent segments (1st use)"** (segments are congruent if they have the same length)

### Completed Reasoning for Each Statement:

1. **Reason: Given**
2. **Reason: Definition of midpoint**
3. **Reason: Given**
4. **Reason: Segment Addition Property**
5. **Reason: Substitution Property (1st use)**
6. **Reason: Substitution Property (2nd use)**
7. **Reason: Simplify**
8. **Reason: Division Property**
9. **Reason: Definition of congruent segments (1st use)**

Would you like further clarification on any part of the proof, or more details on any of the properties used? Here are some related questions that might expand your understanding:

1. Why is the "Segment Addition Property" used in step 4?
2. What is the role of the "Substitution Property" in this proof?
3. How does the definition of midpoint apply to solving this problem?
4. Can you explain how simplifying works in step 7?
5. What is the difference between the "Definition of congruent segments" and the "Substitution Property"?

### Tip:
When working on two-column proofs, it's crucial to match the reasoning steps carefully, ensuring each statement logically follows the previous one using defined geometric properties.

Complete the two-column proof. The statements are provided on the left. Drag the reasons to the corresponding statements. Given: C is the midpoint of AE and BD. AE = BD. Prove: AC ≅ CD.

Learn how to complete a two-column geometric proof to show that two segments are congruent given that a point is the midpoint of two segments and the segments are equal in length. This problem involves understanding the definition of midpoint, segment addition property, and properties of equality, suitable for students in Grades 9-10.

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