Question

1. 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.

Explanation:

Step1: Recall perpendicular - parallel line properties

In a plane, if two lines are perpendicular to the same line, they are parallel to each other. Also, by the definition of perpendicular lines, if a line is perpendicular to another line, the angle between them is 90 degrees.

Step2: Analyze each option

• Option A: Lines $\ell$ and $m$ are perpendicular. This is false because if $\ell\perp n$ and $m\perp n$, then $\ell\parallel m$.
• Option B: Lines $\ell$ and $n$ are perpendicular. Given $\ell\perp n$, this is true.
• Option C: Lines $m$ and $n$ are perpendicular. Given $m\perp n$, this is true.
• Option D: Lines $\ell$ and $m$ are parallel. Since $\ell\perp n$ and $m\perp n$, this is true.
• Option E: Lines $\ell$ and $n$ are parallel. Since $\ell\perp n$, this is false.
• Option F: Lines $m$ and $n$ are parallel. Since $m\perp n$, this is false.

Answer:

B. Lines $\ell$ and $n$ are perpendicular.
C. Lines $m$ and $n$ are perpendicular.
D. Lines $\ell$ and $m$ are parallel.