Parallel Lines and Proofs worksheet for geometry practice.

Now I can see the second proof clearly. This one has a different diagram with three lines intersecting, creating angles labeled 1, 2, 3, and 4.

The given information is: ∠3 ≅ ∠4
We need to prove: ∠1 ≅ ∠2

Looking at the table:
- Statement 1: ∠3 ≅ ∠4 (Given)
- Statement 2: ∠3 ≅ ∠1 (This appears to be based on vertical angles)
- Statement 3: ? (Missing statement)
- Statement 4: ∠1 ≅ ∠2 (Conclusion)

The reason for statement 2 is likely "Vertical Angles Theorem" since ∠3 and ∠1 appear to be vertical angles in the diagram.

For statement 3, since we have ∠3 ≅ ∠4 (given) and ∠3 ≅ ∠1 (vertical angles), we can use the Transitive Property of Congruence to say ∠1 ≅ ∠4.

Then for statement 4, since ∠1 and ∠2 are also vertical angles, we would use the Vertical Angles Theorem again.

Let me now fill in the complete solutions for both proofs.

For the first proof:
1. ∠1 ≅ ∠3 (Given)
2. m∠1 + m∠2 = 180° (Linear Pair Postulate - angles 1 and 2 form a straight line)
3. m∠2 + m∠3 = 180° (Substitution - since ∠1 ≅ ∠3, we can substitute ∠3 for ∠1)

For the second proof:
1. ∠3 ≅ ∠4 (Given)
2. ∠3 ≅ ∠1 (Vertical Angles Theorem)
3. ∠1 ≅ ∠4 (Transitive Property of Congruence)
4. ∠1 ≅ ∠2 (Vertical Angles Theorem - since ∠1 and ∠2 are vertical angles)

Actually, let me reconsider the second proof. If we're trying to prove ∠1 ≅ ∠2, and we know ∠3 ≅ ∠4, and from the diagram it appears that ∠3 and ∠1 are vertical angles, and ∠2 and ∠4 are vertical angles, then:

1. ∠3 ≅ ∠4 (Given)
2. ∠3 ≅ ∠1 (Vertical Angles Theorem)
3. ∠4 ≅ ∠2 (Vertical Angles Theorem)
4. ∠1 ≅ ∠2 (Transitive Property of Congruence - since ∠1 ≅ ∠3, ∠3 ≅ ∠4, and ∠4 ≅ ∠2, then ∠1 ≅ ∠2)

This makes more sense as a logical progression.

Let me provide the final answer with the completed tables.

Based on my analysis, here are the completed proofs:

First Proof:
Given: ∠1 ≅ ∠3
Prove: m∠2 + m∠3 = 180°

| Statement | Reason |
|-----------|--------|
| 1. ∠1 ≅ ∠3 | Given |
| 2. m∠1 + m∠2 = 180° | Linear Pair Postulate (angles 1 and 2 form a straight line) |
| 3. m∠2 + m∠3 = 180° | Substitution (since ∠1 ≅ ∠3, we can substitute ∠3 for ∠1) |

Second Proof:
Given: ∠3 ≅ ∠4
Prove: ∠1 ≅ ∠2

| Statement | Reason |
|-----------|--------|
| 1. ∠3 ≅ ∠4 | Given |
| 2. ∠3 ≅ ∠1 | Vertical Angles Theorem |
| 3. ∠4 ≅ ∠2 | Vertical Angles Theorem |
| 4. ∠1 ≅ ∠2 | Transitive Property of Congruence |

These proofs demonstrate the relationships between angles formed by parallel lines and transversals, as well as the properties of vertical angles.