Parallel line proofs worksheet with four geometry problems and diagrams.

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 (given).
- ∠6 and ∠7 form a linear pair → so they are supplementary.
- Since both ∠4 and ∠6 are supplementary to ∠7, they must be congruent (Congruent Supplements Theorem).
- ∠4 and ∠6 are alternate interior angles.
- If alternate interior angles are congruent, then the lines are parallel → $ j \parallel k $.

✔️ This proof is 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 Converse? Wait — this needs correction. |

Wait — let’s look carefully.

Actually, ∠1 and ∠2 are vertical angles, not alternate interior.

But here’s the issue: Statement 3 says ∠1 ≅ ∠2, but the reason is listed as "Alternate Interior Angles Converse", which is wrong.

Let’s fix it:

| Statements | Reasons |
|-----------|--------|
| 1. $ a \parallel b $ | 1. Given |
| 2. ∠2 ≅ ∠3 | 2. Given |
| 3. ∠1 ≅ ∠2 | 3. Vertical Angles Theorem |
| 4. ∠1 ≅ ∠3 | 4. Transitive Property of Congruence |

✔ Corrected Reason for Step 3: Vertical Angles Theorem
(Vertical angles are always congruent.)

Then Step 4 uses transitive property: if ∠1 ≅ ∠2 and ∠2 ≅ ∠3, then ∠1 ≅ ∠3.

✔️ So the final answer is correct, but the reason for Step 3 is wrong — should be Vertical Angles Theorem, not "Alternate Interior Angles Converse".

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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 |
| 3. ∠4 ≅ ∠8 | 3. Corresponding Angles |

✔ Explanation:
- ∠1 and ∠7 are alternate exterior angles.
- If alternate exterior angles are congruent, then the lines are parallel → $ l \parallel m $
- Now, ∠4 and ∠8 are corresponding angles (same position on each line).
- Since $ l \parallel m $, corresponding angles are congruent → ∠4 ≅ ∠8

✔️ This proof is 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 |
| 3. ∠3 ≅ ∠6 | 3. Alternate Interior Angles Theorem |

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

⚠️ Note: The term used is "Consecutive Exterior Angles Converse" — that’s acceptable, though sometimes called "Same-Side Exterior Angles Converse".

Also, the last statement should be "∠3 ≅ ∠6", and the reason is Alternate Interior Angles Theorem (not converse — we’re using the theorem, not the converse).

So the proof is correct.

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✔ Final Summary of Corrections:

| Problem | Issue | Correction |
|--------|-------|------------|
| 2 | Reason for ∠1 ≅ ∠2 is wrong | Change from "Alternate Interior Angles Converse" to Vertical Angles Theorem |

All other steps are correct.

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✔ Final Answer:

The only error is in Problem 2, Step 3.

Corrected Proof for Problem 2:

| Statements | Reasons |
|-----------|--------|
| 1. $ a \parallel b $ | Given |
| 2. ∠2 ≅ ∠3 | Given |
| 3. ∠1 ≅ ∠2 | Vertical Angles Theorem |
| 4. ∠1 ≅ ∠3 | Transitive Property of Congruence |

All other proofs are correctly completed.

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