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In the given diagram, we are tasked with proving that \( \angle 1 \cong \angle 5 \) based on the provided information and without using the Transitive Property of parallel lines directly.

### Given Information:
1. \( \overline{CB} \parallel \overline{DF} \)
2. \( \overline{CD} \parallel \overline{GE} \)

### Steps for Proof:

1. **Identify Corresponding Angles (Step 1):**
   Since \( \overline{CB} \parallel \overline{DF} \) and \( \overline{CD} \) is a transversal, by the Corresponding Angles Postulate, we know that:
   \[
   \angle 1 \cong \angle 2
   \]
   (Corresponding angles formed by a transversal across parallel lines.)

2. **Identify Corresponding Angles (Step 2):**
   Similarly, since \( \overline{CD} \parallel \overline{GE} \) and \( \overline{DF} \) is the transversal, by the Corresponding Angles Postulate, we can conclude that:
   \[
   \angle 5 \cong \angle 6
   \]
   (Again, corresponding angles formed by the transversal.)

3. **Vertical Angles (Step 3):**
   Now, observe that \( \angle 2 \cong \angle 6 \) because they are vertical angles formed by the intersection of lines \( \overline{CD} \) and \( \overline{GE} \). Vertical angles are always congruent.

4. **Transitivity (Step 4):**
   Using the Transitive Property of Congruence (allowed for angles):
   - From Step 1, \( \angle 1 \cong \angle 2 \)
   - From Step 3, \( \angle 2 \cong \angle 6 \)
   - From Step 2, \( \angle 6 \cong \angle 5 \)

   Therefore, by transitivity, we conclude:
   \[
   \angle 1 \cong \angle 5
   \]

This completes the proof that \( \angle 1 \cong \angle 5 \).

Would you like more details on any part of this proof? Here are five related questions:

1. What is the Corresponding Angles Postulate and how does it apply here?
2. Can you explain why vertical angles are congruent?
3. What other properties of parallel lines could be used to prove congruence of angles?
4. What is the Transitive Property of Congruence, and how is it different from the Transitive Property for parallel lines?
5. How would the proof change if the parallel lines were not given?

**Tip:** When dealing with parallel lines, always look for corresponding, alternate interior, or alternate exterior angles formed by transversals to help identify congruent angles.

Use the given information to prove that ∠1 ≅ ∠5. You are not allowed to use the Transitive Property for parallel lines.

Learn how to prove that ∠1 is congruent to ∠5 in a geometry problem involving parallel lines and transversals. This step-by-step solution uses the Corresponding Angles Postulate and Vertical Angles Theorem, suitable for students in Grades 9-10.

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