two distinct lines, l and m, are each perpendicular to the same line n. select all the true statements. a lines l and m are perpendicular. b lines l and n are perpendicular. c lines m and n are perpendicular. d lines l and m are parallel. e lines l and n are parallel. f lines m and n are parallel.

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# Explanation:
## Step1: Recall the theorem about lines perpendicular to the same line
If two distinct lines are each perpendicular to the same line, then those two lines are parallel to each other. Also, each of those two lines is perpendicular to the third line.

## Step2: Analyze each option
- **Option A**: Lines $ l $ and $ m $ are each perpendicular to line $ n $, so by the theorem, $ l \perp n $ and $ m \perp n $, so this is true.
- **Option B**: Lines $ l $ and $ n $ are perpendicular (given that $ l $ is perpendicular to $ n $), but $ m $ is also perpendicular to $ n $, so $ l $ and $ n $ are not perpendicular to each other in the sense of the option (wait, no: $ l $ is perpendicular to $ n $, so this statement "Lines $ l $ and $ n $ are perpendicular" is true? Wait, the problem says "two distinct lines, $ l $ and $ m $, are each perpendicular to the same line $ n $". So $ l \perp n $ and $ m \perp n $. So:
  - A: $ l \perp m $? No, wait, if two lines are perpendicular to the same line, they are parallel, not perpendicular. Wait, I made a mistake. Let's correct: The theorem is: In a plane, if two lines are perpendicular to the same line, then they are parallel to each other. So:
    - $ l \perp n $ and $ m \perp n $ (given)
    - So $ l \parallel m $ (since both perpendicular to $ n $)
    - Also, $ l \perp n $ (true, given), $ m \perp n $ (true, given)
    - Now let's check each option:
      - A: Lines $ l $ and $ m $ are perpendicular. No, they are parallel. So A is false.
      - B: Lines $ l $ and $ n $ are perpendicular. Yes, because $ l $ is perpendicular to $ n $ (given). So B is true.
      - C: Lines $ m $ and $ n $ are perpendicular. Yes, because $ m $ is perpendicular to $ n $ (given). So C is true.
      - D: Lines $ l $ and $ m $ are parallel. Yes, by the theorem. So D is true.
      - E: Lines $ l $ and $ n $ are parallel. No, $ l $ is perpendicular to $ n $, so they are not parallel. So E is false.
      - F: Lines $ m $ and $ n $ are parallel. No, $ m $ is perpendicular to $ n $, so they are not parallel. So F is false.

Wait, let's re-express the given: "Two distinct lines, $ l $ and $ m $, are each perpendicular to the same line $ n $". So:

- $ l \perp n $ (true)
- $ m \perp n $ (true)
- Then, in a plane, if two lines are perpendicular to the same line, they are parallel. So $ l \parallel m $ (true)

Now check each option:

- A: $ l \perp m $? No, $ l \parallel m $, so A is false.
- B: $ l \perp n $? Yes, given, so B is true.
- C: $ m \perp n $? Yes, given, so C is true.
- D: $ l \parallel m $? Yes, by the theorem, so D is true.
- E: $ l \parallel n $? No, $ l \perp n $, so E is false.
- F: $ m \parallel n $? No, $ m \perp n $, so F is false.

So the true statements are B, C, D.

Wait, but let's confirm the theorem: In Euclidean geometry, if two lines are perpendicular to the same line, then they are parallel to each other. So $ l $ and $ m $ are parallel (D is true). Also, $ l $ is perpendicular to $ n $ (B is true), $ m $ is perpendicular to $ n $ (C is true). So B, C, D are true.

# Answer:
B. Lines $ l $ and $ n $ are perpendicular, C. Lines $ m $ and $ n $ are perpendicular, D. Lines $ l $ and $ m $ are parallel

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Question

Explanation:

Step1: Recall the theorem about lines perpendicular to the same line

Step2: Analyze each option

Answer: