for each angle relationship tell if it proves the lines l and m are parallel. options: angle 2 and 4 are congruent, angles 2 and 6 are congruent, angles 5 and 8 are complementary, angles 6 and 8 are congruent. columns: proves they are parallel, does not prove lines are parallel.

Question

Accepted Answer

To solve this, we analyze each angle pair:

### 1. Angle 2 and 4
These are vertical angles (formed by intersecting lines). Vertical angles are always congruent, but this does not relate to the parallelism of lines $ l $ and $ m $ (since vertical angles exist regardless of parallel lines). So this pair **does not prove $ l \parallel m $**.

### 2. Angles 2 and 6
Angle 2 and angle 6 are corresponding angles (if lines $ l $ and $ m $ are cut by a transversal, corresponding angles being congruent is a postulate for parallel lines). Wait, no—wait, let’s re-examine the diagram. Wait, actually, angle 2 and angle 6: if $ l $ and $ m $ are parallel, corresponding angles are congruent. But wait, no—wait, the transversal here: let's see, lines $ l $ and $ m $, and the other lines. Wait, no, angle 2 and angle 6: if they are congruent, by the Corresponding Angles Postulate, lines $ l $ and $ m $ would be parallel. Wait, no, maybe I mislabeled. Wait, no—wait, the first pair: angle 2 and 4 are vertical angles (so congruent, but not about parallelism). The second pair: angles 2 and 6—if they are congruent, that’s corresponding angles, so that would prove $ l \parallel m $. Wait, but the options are “proves they are parallel” or “does not prove”. Wait, let's clarify:

- **Angle 2 and 4**: Vertical angles (congruent by vertical angles theorem), but this is about intersecting lines, not parallel lines. So this pair does not prove $ l \parallel m $.
- **Angles 2 and 6**: If congruent, by Corresponding Angles Postulate, $ l \parallel m $ (so this proves parallelism).
- **Angles 5 and 8**: Vertical angles (congruent, but not about parallelism—does not prove $ l \parallel m $).
- **Angles 6 and 8**: Vertical angles? No, angle 6 and 8: wait, angle 6 and 8 are vertical angles? No, angle 5 and 8 are vertical, angle 6 and 7 are vertical. Wait, maybe a typo. Wait, the last option: “Angles 6 and 8 are congruent”—no, angle 6 and 8: angle 5 and 8 are vertical (congruent), angle 6 and 7 are vertical. So angle 6 and 8: if lines are parallel, maybe alternate interior, but no—wait, no, angle 6 and 8: if they are congruent, that would be alternate interior? Wait, no, let's re-express:

Wait, the problem is to determine for each angle relationship if it proves $ l \parallel m $ or not. Let's take each:

1. **Angle 2 and 4 are congruent**: Vertical angles (always congruent, regardless of $ l $ and $ m $ being parallel). So this **does not prove** $ l \parallel m $.
2. **Angles 2 and 6 are congruent**: Corresponding angles. If corresponding angles are congruent, lines are parallel (Corresponding Angles Postulate). So this **proves** $ l \parallel m $.
3. **Angles 5 and 8 are congruent**: Vertical angles (always congruent, regardless of $ l $ and $ m $ being parallel). So this **does not prove** $ l \parallel m $.
4. **Angles 6 and 8 are congruent**: Wait, angle 6 and 8: angle 5 and 8 are vertical (congruent), angle 6 and 7 are vertical. If angle 6 and 8 are congruent, that would mean angle 6 = angle 8, but angle 5 = angle 8 (vertical), so angle 6 = angle 5—alternate interior angles? Wait, no, maybe a mistake. But the key is: vertical angles (like 2&4, 5&8) do not prove parallelism, while corresponding/alternate interior/exterior angles (like 2&6) do.

### For “Angle 2 and 4 are congruent”:
# Brief Explanations:
Angle 2 and 4 are vertical angles (congruent by vertical angles theorem), but vertical angles exist regardless of whether $ l $ and $ m $ are parallel. Thus, this relationship does not prove $ l \parallel m $.
# Answer:
Does not prove lines are parallel

### For “Angles 2 and 6 are congruent”:
# Brief Explanations:
Angles 2 and 6 are corresponding angles. By the Corresponding Angles Postulate, if corresponding angles are congruent, the lines cut by the transversal are parallel. Thus, this relationship proves $ l \parallel m $.
# Answer:
Proves they are parallel

### For “Angles 5 and 8 are congruent”:
# Brief Explanations:
Angles 5 and 8 are vertical angles (congruent by vertical angles theorem), but vertical angles exist regardless of whether $ l $ and $ m $ are parallel. Thus, this relationship does not prove $ l \parallel m $.
# Answer:
Does not prove lines are parallel

### For “Angles 6 and 8 are congruent”:
Wait, angle 6 and 8: angle 5 and 8 are vertical (congruent), so if angle 6 = angle 8, then angle 6 = angle 5 (alternate interior angles). By Alternate Interior Angles Theorem, if alternate interior angles are congruent, lines are parallel. Wait, maybe a mislabel. But assuming angle 6 and 8: if congruent, and angle 5 and 8 are vertical (so angle 5 = angle 8), then angle 6 = angle 5 (alternate interior), so lines $ l \parallel m $. But likely, the intended is: vertical angles (like 2&4, 5&8) do not prove, while corresponding/alternate (like 2&6, 6&8 if alternate) do. But based on standard problems, vertical angles (2&4, 5&8) do not prove parallelism, while corresponding (2&6) do.

(Note: The exact answer depends on the specific angle pair, but the key is distinguishing vertical angles (no parallelism proof) from corresponding/alternate angles (parallelism proof).)

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QUESTION IMAGE

Question

Explanation:

1. Angle 2 and 4

2. Angles 2 and 6

For “Angle 2 and 4 are congruent”:

Answer:

For “Angles 2 and 6 are congruent”: