Question

use the figure to identify the following statements (14 - 17).

1. find a line that is skew to \\( \overleftrightarrow { c d } \\)
2. find four lines parallel to \\( \overleftrightarrow { d y } \\)
3. how many lines are skew to \\( \overleftrightarrow { z v } \\)
4. how many lines intersect \\( \overleftrightarrow { b w } \\) and list which lines they are.

Explanation:

13. Skew Lines

Skew lines are non - parallel and non - intersecting lines in 3D space.
For \(\overleftrightarrow{CD}\), \(\overleftrightarrow{AZ}\) is skew.

14. Parallel Lines

Parallel lines have the same direction.
If we assume the figure is a prism - like structure (based on the naming of vertices), lines parallel to \(\overleftrightarrow{DY}\) are \(\overleftrightarrow{EX}\), \(\overleftrightarrow{AB}\), \(\overleftrightarrow{ZW}\), \(\overleftrightarrow{VC}\)

15. Counting Skew Lines

First, count all the lines in the figure. Let's assume there are \(n\) lines. Lines that are parallel or intersecting with \(\overleftrightarrow{ZV}\) are excluded.
If we assume there are 12 lines in total (by counting all the edge - representing lines in the 3D figure). Lines parallel to \(\overleftrightarrow{ZV}\) are \(\overleftrightarrow{WC}\), \(\overleftrightarrow{AY}\), \(\overleftrightarrow{DX}\). Lines intersecting \(\overleftrightarrow{ZV}\) are \(\overleftrightarrow{ZE}\), \(\overleftrightarrow{ZV}\) itself (but we don't count it), \(\overleftrightarrow{VA}\), \(\overleftrightarrow{VD}\). So the number of skew lines is \(4\)

16. Intersecting Lines

Lines that intersect \(\overleftrightarrow{BW}\) are \(\overleftrightarrow{AB}\), \(\overleftrightarrow{BC}\), \(\overleftrightarrow{BW}\) itself (but we exclude it), \(\overleftrightarrow{DW}\), \(\overleftrightarrow{VW}\). So there are \(4\) lines \(\overleftrightarrow{AB}\), \(\overleftrightarrow{BC}\), \(\overleftrightarrow{DW}\), \(\overleftrightarrow{VW}\)

Answer:

1. \(\overleftrightarrow{AZ}\)
2. \(\overleftrightarrow{EX}\), \(\overleftrightarrow{AB}\), \(\overleftrightarrow{ZW}\), \(\overleftrightarrow{VC}\)
3. \(4\)
4. \(4\) lines: \(\overleftrightarrow{AB}\), \(\overleftrightarrow{BC}\), \(\overleftrightarrow{DW}\), \(\overleftrightarrow{VW}\)