https://file.aiquickdraw.com/mathbot/file/1726616618689cmxd7otf.jpg

The image shows a geometric proof where the objective is to prove that lines \( c \parallel d \), given that \( a \parallel b \) and \( \angle 1 \cong \angle 3 \).

### Proof Structure:
- **Given:**
  1. \( a \parallel b \)
  2. \( \angle 1 \cong \angle 3 \)

- **To Prove:**
  \( c \parallel d \)

- **Steps:**
  1. **Statement**: \( a \parallel b \) and \( \angle 1 \cong \angle 3 \)  
     **Reason**: Given
     
  2. **Statement**: \( \angle 1 \cong \angle 2 \)  
     **Reason**: Alternate interior angle theorem (since \( a \parallel b \))
     
  3. **Statement**: \( \angle 2 \cong \angle 3 \)  
     **Reason**: Alternate exterior angle theorem (converse). Since \( \angle 1 \cong \angle 3 \), by transitivity, \( \angle 2 \cong \angle 3 \).
     
  4. **Statement**: \( c \parallel d \)  
     **Reason**: Transitive property (since \( \angle 2 \cong \angle 3 \), the lines \( c \) and \( d \) must be parallel by the converse of the alternate exterior angle theorem).

Would you like more details or clarifications on this proof?

Here are some related questions:
1. What are alternate interior angles?
2. How does the alternate exterior angle theorem apply in parallel line proofs?
3. Can you explain the transitive property in geometry?
4. What is the difference between alternate interior and alternate exterior angles?
5. What is the converse of a theorem, and why is it important in proofs?

**Tip**: When working with parallel lines and angles, always check for corresponding or alternate angles to establish relationships between lines.

Prove that lines c are parallel to d, given that lines a are parallel to b, and angles 1 and 3 are congruent.

In this geometric proof, we demonstrate how to prove that lines c are parallel to d by using the alternate interior and exterior angle theorems, given that lines a are parallel to b and angles 1 and 3 are congruent. This step-by-step solution is suitable for students in Grades 8-10.

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