Parallel Line Proofs - all filled in - PARALLEL LINE Proofs 1 ... - Free Printable

Let's go through each of the parallel line proofs step by step and correct or complete the missing parts in the proof tables. We'll also explain the reasoning behind each step.

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Problem 1

Given: ∠4 and ∠7 are supplementary
Prove: $ j \parallel k $

#### Proof Table:

| Statements | Reasons |
|-----------|--------|
| 1. ∠4 and ∠7 are supplementary | 1. Given |
| 2. ∠6 and ∠7 are supplementary | 2. Linear pair (they form a straight line) |
| 3. ∠4 ≅ ∠6 | 3. Congruent Supplements Theorem (If two angles are supplementary to the same angle, they are congruent) |
| 4. $ j \parallel k $ | 4. Alternate Interior Angles Converse (If alternate interior angles are congruent, then lines are parallel) |

✔ Explanation:
- ∠4 and ∠7 are supplementary → sum to 180°.
- ∠6 and ∠7 form a linear pair → also supplementary → sum to 180°.
- So both ∠4 and ∠6 are supplementary to ∠7 → they must be congruent.
- ∠4 and ∠6 are alternate interior angles → if they're congruent, the lines $ j $ and $ k $ are parallel.

✔️ Correct!

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Problem 2

Given: $ a \parallel b $, ∠2 ≅ ∠3
Prove: ∠1 ≅ ∠3

#### Proof Table:

| Statements | Reasons |
|-----------|--------|
| 1. $ a \parallel b $ | 1. Given |
| 2. ∠2 ≅ ∠3 | 2. Given |
| 3. ∠1 ≅ ∠2 | 3. Alternate Interior Angles Theorem (since $ a \parallel b $, ∠1 and ∠2 are alternate interior angles) |
| 4. ∠1 ≅ ∠3 | 4. Transitive Property of Congruence (∠1 ≅ ∠2 and ∠2 ≅ ∠3 ⇒ ∠1 ≅ ∠3) |

✔ Explanation:
- Since $ a \parallel b $, and a transversal cuts them, ∠1 and ∠2 are alternate interior angles, so they are congruent.
- Given ∠2 ≅ ∠3.
- Therefore, ∠1 ≅ ∠3 by substitution/transitivity.

✔️ Correct!

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Problem 3

Given: ∠1 ≅ ∠7
Prove: ∠4 ≅ ∠8

#### Proof Table:

| Statements | Reasons |
|-----------|--------|
| 1. ∠1 ≅ ∠7 | 1. Given |
| 2. $ l \parallel m $ | 2. Alternate Exterior Angles Converse (if alternate exterior angles are congruent, lines are parallel) |
| 3. ∠4 ≅ ∠8 | 3. Corresponding Angles Postulate (since $ l \parallel m $, corresponding angles are congruent) |

✔ Explanation:
- ∠1 and ∠7 are alternate exterior angles → if they are congruent, then lines $ l $ and $ m $ are parallel.
- Once we know $ l \parallel m $, then ∠4 and ∠8 are corresponding angles → therefore congruent.

✔️ Correct!

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Problem 4

Given: ∠5 and ∠8 are supplementary
Prove: ∠3 ≅ ∠6

#### Proof Table:

| Statements | Reasons |
|-----------|--------|
| 1. ∠5 and ∠8 are supplementary | 1. Given |
| 2. $ r \parallel s $ | 2. Consecutive Exterior Angles Converse (if consecutive exterior angles are supplementary, lines are parallel) |
| 3. ∠3 ≅ ∠6 | 3. Alternate Interior Angles Theorem (since $ r \parallel s $, alternate interior angles are congruent) |

✔ Explanation:
- ∠5 and ∠8 are consecutive exterior angles (also called same-side exterior angles).
- If they are supplementary, then the lines $ r $ and $ s $ are parallel.
- Now, ∠3 and ∠6 are alternate interior angles → when lines are parallel, these are congruent.

✔️ Correct!

> Note: The term "consecutive exterior angles converse" is sometimes called Same-Side Exterior Angles Converse — it’s valid.

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✔ Final Summary:

All four proofs are correctly filled out with appropriate statements and reasons. Here’s a quick recap:

| Problem | Key Concept Used |
|--------|------------------|
| 1 | Supplementary angles → alternate interior angles → parallel lines |
| 2 | Parallel lines → alternate interior angles + transitive property |
| 3 | Alternate exterior angles → parallel lines → corresponding angles |
| 4 | Same-side exterior angles supplementary → parallel lines → alternate interior angles |

These proofs rely on:
- Angle relationships (linear pairs, vertical angles, etc.)
- Parallel line theorems and their converses
- Transitive property of congruence

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✔ Answer: All proofs are correct as written. No corrections needed.