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The problem asks to determine which lines are parallel based on the given angle relationships. To solve this, we need to apply properties of parallel lines and angle pairs (e.g., corresponding angles, alternate interior angles, and consecutive interior angles) formed by transversals.

### Let's go through each angle relationship one by one:

1. **∠13 ≅ ∠17**:  
   - These are alternate interior angles. If they are congruent, then the lines cut by the transversal must be parallel.
   - **Conclusion**: This shows that **a ∥ b** (Option 1).

2. **∠8 ≅ ∠19**:  
   - These are alternate interior angles as well. If these angles are congruent, then the lines they cut must be parallel.
   - **Conclusion**: This shows that **b ∥ c** (Option 2).

3. **∠10 ≅ ∠23**:  
   - These are alternate interior angles, and if they are congruent, then the lines cut by the transversal are parallel.
   - **Conclusion**: This shows that **a ∥ c** (Option 3).

4. **m∠20 + m∠21 = 180°**:  
   - These angles are consecutive interior angles. If their sum is 180°, the lines they intersect must be parallel.
   - **Conclusion**: This shows that **d ∥ e** (Option 4).

5. **m∠14 + m∠17 = 180°**:  
   - These angles don't correspond to consecutive interior or other angles that can prove parallelism. Therefore, this doesn't prove any lines parallel.
   - **Conclusion**: **Doesn't prove lines are parallel** (Option 5).

6. **∠4 ≅ ∠9**:  
   - These are alternate interior angles. If they are congruent, the lines cut by the transversal are parallel.
   - **Conclusion**: This shows that **b ∥ c** (Option 2).

### Final Answers:
- ∠13 ≅ ∠17: **1 (a ∥ b)**
- ∠8 ≅ ∠19: **2 (b ∥ c)**
- ∠10 ≅ ∠23: **3 (a ∥ c)**
- m∠20 + m∠21 = 180°: **4 (d ∥ e)**
- m∠14 + m∠17 = 180°: **5 (Doesn't prove lines are parallel)**
- ∠4 ≅ ∠9: **2 (b ∥ c)**

Let me know if you'd like more details or if you have any other questions!

### Related questions:
1. What are alternate interior angles?
2. How do corresponding angles indicate parallelism?
3. What are consecutive interior angles?
4. How do we know when angles form a transversal?
5. Why can't certain angle relationships prove parallelism?

### Tip:
Remember that alternate interior angles and corresponding angles are key to identifying parallel lines when cut by a transversal.

Determine which lines, if any, are parallel. Choices may be repeated and some might not be used at all. Based on given angle relationships and choices provided, identify parallel lines.

Learn how to determine parallel lines using alternate interior angles, corresponding angles, and consecutive interior angles theorems. This geometry problem is suitable for students in Grades 8-10.

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