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The question asks for completing the proof of a geometric statement about a pair of vertical angles, specifically related to segments.

The diagram shows two intersecting lines, creating four angles at the intersection. The goal is to complete the proof steps.

Here are some general steps for proving a geometric statement about segments:

1. **Identify the given information:** The problem seems to state that \( C \) is the midpoint of segments \( AE \) and \( BD \). This means that the two segments are bisected at point \( C \).

2. **Set up the equation:** The equality of segment lengths will follow from the definition of the midpoint. For example, \( AC = CE \) and \( BC = CD \).

3. **Use properties:** Use segment addition, substitution, and the definition of the midpoint to establish the equality of different parts of the segments.

The blocks on the side suggest different types of reasoning, such as definitions (e.g., midpoint), properties (e.g., substitution, division), and basic steps like simplification.

To assist you in completing the proof, please provide more specific details or steps of the proof, and I can help you arrange them correctly.

Would you like to provide more information, or do you have specific questions? Here are five questions to guide you further:

1. What exact segments are being compared or proved equal in this proof?
2. What is the role of midpoint \( C \) in the proof?
3. Can you identify which step might involve the definition of the midpoint?
4. How does substitution play a role in this proof process?
5. What might be the final conclusion that the proof is aiming to establish?

**Tip:** When working on proofs, it is often helpful to break down each statement and reason why it must be true based on prior knowledge or given information.

The query submitted by the user involves completing a proof of a geometric statement about intersecting segments.

Learn how to complete a geometric proof involving intersecting lines, vertical angles, and the midpoint of segments. This problem explores key geometric concepts such as segment addition, the midpoint theorem, and substitution property. Suitable for students in Grades 8-10.

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