Geometry homework worksheet on properties of parallel lines and proofs, including alternate exterior angles and same-side exterior angles.

Let’s solve both proofs step-by-step, filling in the missing statements and reasons.

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## Proof #1: Proof of Alternate Exterior Angles congruent

Given:
Line *l* is parallel to line *m* (l ∥ m)
Transversal *t* intersects them, forming angles 1, 2, and 3 as shown.
Angle 1 and angle 3 are alternate exterior angles.

Prove:
∠1 ≅ ∠3

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Completed Proof Table:

| Statements | Reasons |
|----------------------------------|--------------------------------------------------|
| 1. l ∥ m | 1. Given |
| 2. ∠1 ≅ ∠2 | 2. Vertical Angles Theorem |
| 3. ∠2 ≅ ∠3 | 3. If lines are parallel, then corresponding angles are congruent. |
| 4. ∠1 ≅ ∠3 | 4. Transitive Property of Congruence |

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✔ Explanation:

- Step 1: We start with what’s given — the two lines are parallel.
- Step 2: ∠1 and ∠2 are vertical angles (they form an “X” at the intersection). Vertical angles are always congruent — that’s a basic theorem.
- Step 3: ∠2 and ∠3 are corresponding angles (both on the same side of the transversal, one above line *l*, one above line *m*). Since *l* ∥ *m*, corresponding angles are congruent.
- Step 4: Since ∠1 ≅ ∠2 and ∠2 ≅ ∠3, by the Transitive Property, ∠1 ≅ ∠3.

✔ So, alternate exterior angles (∠1 and ∠3) are congruent when lines are parallel.

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## Proof #2: Proof of Same-Side Exterior Angles supplementary

Given:
l ∥ m
Transversal *t* forms angles 1, 2, and 3 as shown.
∠1 and ∠3 are same-side exterior angles (on the same side of the transversal, outside the parallel lines).

Prove:
∠1 is supplementary to ∠3 → meaning m∠1 + m∠3 = 180°

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Completed Proof Table:

| Statements | Reasons |
|-------------------------------------------|--------------------------------------------------|
| 1. l ∥ m | 1. Given |
| 2. m∠1 + m∠2 = 180 | 2. Angle Addition Postulate |
| 3. m∠2 = m∠3; ∠2 ≅ ∠3 | 3. If lines are parallel, then corresponding angles are congruent. |
| 4. m∠1 + m∠3 = 180 | 4. Substitution Property |
| 5. ∠1 is supplementary to ∠3 | 5. Definition of Supplementary Angles |

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✔ Explanation:

- Step 1: Given — lines are parallel.
- Step 2: ∠1 and ∠2 are adjacent angles that form a straight line (linear pair), so their measures add up to 180° — this is the Angle Addition Postulate (or Linear Pair Postulate).
- Step 3: ∠2 and ∠3 are corresponding angles (same position relative to the parallel lines and transversal). Since l ∥ m, they are congruent → so their measures are equal.
- Step 4: Substitute m∠3 for m∠2 in Step 2 → so m∠1 + m∠3 = 180.
- Step 5: By definition, if two angles add up to 180°, they are supplementary.

✔ Therefore, same-side exterior angles (∠1 and ∠3) are supplementary when lines are parallel.

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## 🎯 Final Answer Summary:

Proof #1:

- Statements:
1. l ∥ m
2. ∠1 ≅ ∠2
3. ∠2 ≅ ∠3
4. ∠1 ≅ ∠3
- Reasons:
1. Given
2. Vertical Angles Theorem
3. Corresponding Angles Postulate (parallel lines)
4. Transitive Property of Congruence

Proof #2:

- Statements:
1. l ∥ m
2. m∠1 + m∠2 = 180
3. m∠2 = m∠3; ∠2 ≅ ∠3
4. m∠1 + m∠3 = 180
5. ∠1 is supplementary to ∠3
- Reasons:
1. Given
2. Angle Addition Postulate
3. Corresponding Angles Postulate (parallel lines)
4. Substitution Property
5. Definition of Supplementary Angles

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Let me know if you’d like a diagram explanation or want to extend this to other angle pairs!