3.3-3.4 Proving Parallel Lines | PDF | Elementary Geometry | Geometry - Free Printable

Let’s solve each proof step by step, filling in the missing Statements and Reasons using the given information and geometric theorems.

---

## 🔹 Proof 1: Given ℓ ∥ m, ∠1 ≅ ∠3 → Prove r ∥ p

We are told that lines ℓ and m are parallel, and angle 1 is congruent to angle 3. We need to prove that lines r and p are parallel.

Key Idea:

Use corresponding angles or alternate exterior angles to show that r ∥ p.

Looking at the diagram (implied), ∠1 and ∠2 are likely vertical angles, so they’re congruent. Since ∠1 ≅ ∠3 (given), then ∠2 ≅ ∠3 by substitution. If ∠2 and ∠3 are corresponding angles or alternate exterior angles formed by transversal t cutting lines r and p, then r ∥ p by the Converse of the Corresponding Angles Postulate or Alternate Exterior Angles Theorem.

---

Completed Proof Table:

| Statement | Reason |
|-----------------------|-------------------------------|
| 1. ℓ ∥ m | 1. Given |
| 2. ∠1 ≅ ∠2 | 2. Vertical Angles Theorem |
| 3. ∠1 ≅ ∠3 | 3. Given |
| 4. ∠2 ≅ ∠3 | 4. Substitution Property of Equality |
| 5. r ∥ p | 5. Alternate Exterior Angles Converse (or Corresponding Angles Converse) |

> ✔ Note: In the Word Bank, “Alt. Ext. ∠” is provided — this suggests we’re using Alternate Exterior Angles. So if ∠2 and ∠3 are alternate exterior angles and congruent, then r ∥ p.

---

## 🔹 Proof 2: Given p ∥ q → Prove m∠1 = m∠5

This is a classic Corresponding Angles Theorem.

When two parallel lines are cut by a transversal, corresponding angles are congruent.

In the diagram, ∠1 and ∠5 are corresponding angles (both in the upper-left position relative to their respective lines).

---

Completed Proof Table:

| Statements | Reasons |
|--------------------|---------------------------------|
| 1. p ∥ q | 1. Given |
| 2. m∠3 = m∠5 | 2. Corresponding Angles Theorem (p ∥ q, so ∠3 ≅ ∠5) |
| 3. m∠1 = m∠3 | 3. Vertical Angles Theorem (∠1 and ∠3 are vertical) |
| 4. m∠1 = m∠5 | 4. Transitive Property of Equality (or Substitution) |

> ✔ Note: The word bank includes “Vertical ∠” and “Corr. ∠”, which confirms these reasons.

---

## 🔹 Proof 3: Alternate Interior Angles Theorem — Given p ∥ q, ∠5 and ∠6 are linear pair → Prove m∠1 = m∠5

This is proving that alternate interior angles are congruent when lines are parallel.

We are told ∠5 and ∠6 form a linear pair, so they are supplementary (sum to 180°). Also, since p ∥ q, consecutive interior angles (∠3 and ∠6) are supplementary.

So we can set up equations and use substitution to show ∠1 ≅ ∠5.

---

Completed Proof Table:

| Statements | Reasons |
|-------------------------------------|----------------------------------------------|
| 1. p ∥ q | 1. Given |
| 2. ∠3 and ∠6 are supplementary | 2. Consecutive Interior Angles Theorem (p ∥ q) |
| 3. m∠3 + m∠6 = 180° | 3. Definition of Supplementary Angles |
| 4. ∠5 and ∠6 are a linear pair | 4. Given |
| 5. ∠5 and ∠6 are supplementary | 5. Linear Pair Theorem |
| 6. m∠5 + m∠6 = 180° | 6. Definition of Supplementary Angles |
| 7. m∠3 + m∠6 = m∠5 + m∠6 | 7. Substitution Property of Equality (from 3 & 6) |
| 8. m∠3 = m∠5 | 8. Subtraction Property of Equality (subtract m∠6 from both sides) |

> ✔ Note: ∠1 and ∠3 are vertical angles, so m∠1 = m∠3. But the proof stops at showing m∠3 = m∠5. To complete the full proof that m∠1 = m∠5, you’d add:
>
> 9. m∠1 = m∠3 → 9. Vertical Angles Theorem
> 10. m∠1 = m∠5 → 10. Substitution (from 8 and 9)
>
> However, the table only goes to statement 8, and the goal was to prove m∠1 = m∠5, so perhaps the diagram implies ∠1 and ∠3 are vertical, and the final step is implied. But strictly speaking, step 8 proves m∠3 = m∠5, and since ∠1 and ∠3 are vertical, m∠1 = m∠3, so m∠1 = m∠5 follows.

But since the problem says “Prove: m∠1 = m∠5” and ends at step 8 with m∠3 = m∠5, it’s likely expecting us to recognize that ∠1 and ∠3 are vertical, so the final conclusion is implied or should be added.

However, based on the structure of the table and the Word Bank, the proof as written ends at step 8, and we’ve shown m∠3 = m∠5 — which, combined with vertical angles, gives m∠1 = m∠5.

---

## ✔ Final Answer Summary

Proof 1:

| Statement | Reason |
|-----------------------|-------------------------------|
| 1. ℓ ∥ m | Given |
| 2. ∠1 ≅ ∠2 | Vertical Angles Theorem |
| 3. ∠1 ≅ ∠3 | Given |
| 4. ∠2 ≅ ∠3 | Substitution Property of Equality |
| 5. r ∥ p | Alternate Exterior Angles Converse |

---

Proof 2:

| Statements | Reasons |
|--------------------|---------------------------------|
| 1. p ∥ q | Given |
| 2. m∠3 = m∠5 | Corresponding Angles Theorem |
| 3. m∠1 = m∠3 | Vertical Angles Theorem |
| 4. m∠1 = m∠5 | Transitive Property of Equality |

---

Proof 3:

| Statements | Reasons |
|-------------------------------------|----------------------------------------------|
| 1. p ∥ q | Given |
| 2. ∠3 and ∠6 are supplementary | Consecutive Interior Angles Theorem |
| 3. m∠3 + m∠6 = 180° | Def. of Supp. Angles |
| 4. ∠5 and ∠6 are a linear pair | Given |
| 5. ∠5 and ∠6 are supplementary | Linear Pair Theorem |
| 6. m∠5 + m∠6 = 180° | Def. of Supp. Angles |
| 7. m∠3 + m∠6 = m∠5 + m∠6 | Substitution Prop. = |
| 8. m∠3 = m∠5 | Subtraction Prop. = |

> 💡 To fully conclude m∠1 = m∠5, add:
> 9. m∠1 = m∠3 → Vertical Angles Theorem
> 10. m∠1 = m∠5 → Substitution (from 8 & 9)

But since the table stops at 8, and the word bank doesn’t include “Vertical ∠” for this section, it may be assumed ∠1 and ∠3 are vertical, and the proof is accepted as-is.

---

✔ All proofs completed using standard geometric theorems and properties.