Parallel Line Proofs Worksheet - Matching Definitions and Completing Angle Proofs

Absolutely! Let’s solve the problems step by step, starting with matching the vocabulary and then completing the proofs.

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## Part 1: Vocabulary Matching

We need to match each definition to the correct term (A–F).

Definitions:

1. Property which states if a = b and b = c, then a = c.
→ This is the Transitive Property.
✔ Answer: C. Transitive Property

2. Two lines that will never intersect.
→ These are Parallel Lines.
✔ Answer: D. Parallel Lines

3. A line that intersects two or more parallel lines.
→ This is a Transversal.
✔ Answer: F. Transversal

4. Two angles which create a straight line together and their measures add up to 180 degrees.
→ These are Supplementary Angles.
✔ Answer: B. Supplementary Angles

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✔ Final Matching:
1. C
2. D
3. F
4. B

*(Note: A. Given and E. Substitution Property are not used in this matching — they’ll be used in the proofs below.)*

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## Part 2: Complete the Proofs about Congruent Angles

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Proof 1:

Given:
a || b
c || d
∠1 ≅ ∠5

Prove:
∠7 ≅ ∠14

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#### Statement & Reason Table:

| Statement | Reason |
|--------------------|----------------------------|
| 1. a || b | 1. Given |
| 2. c || d | 2. Given |
| 3. ∠1 ≅ ∠5 | 3. Given |
| 4. ∠7 ≅ ∠1 | 4. Vertical Angles Theorem *(∠7 and ∠1 are vertical angles)* |
| 5. ∠7 ≅ ∠5 | 5. Transitive Property *(from 3 and 4: ∠7 ≅ ∠1 and ∠1 ≅ ∠5 ⇒ ∠7 ≅ ∠5)* |
| 6. ∠5 ≅ ∠14 | 6. Corresponding Angles Postulate *(since c || d and transversal cuts them, ∠5 and ∠14 are corresponding)* |
| 7. ∠7 ≅ ∠14 | 7. Transitive Property *(from 5 and 6: ∠7 ≅ ∠5 and ∠5 ≅ ∠14 ⇒ ∠7 ≅ ∠14)* |

> 💡 *Note: Although “Vertical Angles Theorem” isn’t listed in your vocabulary box, it’s standard knowledge. If you must use only the given terms, you could say “Given” for statement 4 if it was provided, but since it’s not, we assume it's known geometry. Alternatively, if your curriculum expects “Definition of Vertical Angles”, that’s acceptable too.*

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Proof 2:

Given:
a || b
c || d
∠8 ≅ ∠16

Prove:
∠12 ≅ ∠10

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#### Statement & Reason Table:

| Statement | Reason |
|--------------------|----------------------------|
| 1. a || b | 1. Given |
| 2. c || d | 2. Given |
| 3. ∠8 ≅ ∠16 | 3. Given |
| 4. ∠12 ≅ ∠8 | 4. Vertical Angles Theorem *(∠12 and ∠8 are vertical angles)* |
| 5. ∠12 ≅ ∠16 | 5. Transitive Property *(from 3 and 4: ∠12 ≅ ∠8 and ∠8 ≅ ∠16 ⇒ ∠12 ≅ ∠16)* |
| 6. ∠16 ≅ ∠10 | 6. Alternate Interior Angles Theorem *(since a || b and transversal cuts them, ∠16 and ∠10 are alternate interior angles)* |
| 7. ∠12 ≅ ∠10 | 7. Transitive Property *(from 5 and 6: ∠12 ≅ ∠16 and ∠16 ≅ ∠10 ⇒ ∠12 ≅ ∠10)* |

> 💡 Again, “Vertical Angles Theorem” and “Alternate Interior Angles Theorem” are standard. If restricted to vocabulary list, you may need to adjust — but logically, these are required steps.

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## ✔ Final Answers Summary:

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Vocabulary Matching:

1. C
2. D
3. F
4. B

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Proof 1:

| Statement | Reason |
|--------------------|----------------------------|
| 1. a || b | Given |
| 2. c || d | Given |
| 3. ∠1 ≅ ∠5 | Given |
| 4. ∠7 ≅ ∠1 | Vertical Angles Theorem |
| 5. ∠7 ≅ ∠5 | Transitive Property |
| 6. ∠5 ≅ ∠14 | Corresponding Angles Postulate |
| 7. ∠7 ≅ ∠14 | Transitive Property |

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Proof 2:

| Statement | Reason |
|--------------------|----------------------------|
| 1. a || b | Given |
| 2. c || d | Given |
| 3. ∠8 ≅ ∠16 | Given |
| 4. ∠12 ≅ ∠8 | Vertical Angles Theorem |
| 5. ∠12 ≅ ∠16 | Transitive Property |
| 6. ∠16 ≅ ∠10 | Alternate Interior Angles Theorem |
| 7. ∠12 ≅ ∠10 | Transitive Property |

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Let me know if you’d like this formatted as a printable worksheet or need help labeling the diagram!