https://file.aiquickdraw.com/mathbot/file/1726100525781mwifofbf.jpg

The problem in the image is a proof involving angle congruence, and the question is asking for the missing reason in part c. To determine the correct reason, let's analyze the proof step by step:

1. **Given: \(\angle 1 \cong \angle 3\)**  
   **Reason**: Given.
  
2. **Given: \(\angle 3 \cong \angle 6\)**  
   **Reason**: Given.

3. **\(\angle 1 \cong \angle 6\)**  
   **Reason**: This uses the **Transitive Property of Congruence** (which states that if \(\angle 1 \cong \angle 3\) and \(\angle 3 \cong \angle 6\), then \(\angle 1 \cong \angle 6\)).

4. **\(\angle 1 \cong \angle 4\)**  
   **Reason**: Here, the statement is missing the reason. From the diagram, angles 1 and 4 are vertical angles (angles that are opposite each other when two lines intersect). Vertical angles are always congruent, so the missing reason should be **Vertical Angles**.

5. **\(\angle 6 \cong \angle 4\)**  
   **Reason**: This uses the **Transitive Property of Congruence** again, because \(\angle 1 \cong \angle 6\) and \(\angle 1 \cong \angle 4\), so \(\angle 6 \cong \angle 4\).

Given these details, the correct reason for the missing reason in part c is **Vertical Angles**.

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Would you like further details or have any questions? Here are five related questions you might find interesting:

1. What is the definition of the Transitive Property of Congruence?
2. How are vertical angles formed?
3. Why are vertical angles always congruent?
4. Can the Transitive Property be used with segments as well as angles?
5. How can you identify linear pairs of angles in a diagram?

**Tip:** Remember, the Transitive Property of Congruence is a powerful tool in geometric proofs, allowing you to link multiple congruences together logically.

What is the reason for the missing reason in part c?

In this geometry proof, find the missing reason for congruence between angles. Learn how the Transitive Property of Congruence and Vertical Angles Theorem apply to solve this problem, suitable for Grades 8-10.

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