Exercise 24 Page 130

a Let's extend the line that passes through points

N

and

Q,

N Q .

From the figure, we can see that $N Q$ is a vertical line along one of the sides of the cube. Any line parallel to this line must also be vertical. We can find these lines along the remaining edges of the cube.

There are three parallel lines: $R K,$ $P M,$ and $S L .$

b Let's extend all of the lines that intersect

N Q .

Notice that each one will pass through either

N

Q .

There are four lines that intersect $N Q :$ $K N,$ $P Q,$ $R Q,$ and $M N .$

c Lines that are skew to

N Q

are neither coplanar with

N Q,

nor do they intersect

N Q .

Examining the figure, we can determine that

N Q

is a part of two separate planes,

N Q R

and

N Q P .

By the definition of skew, we can rule out any line that runs along either of these planes.

P Q \times M N \times P M \times K N \times R Q \times K R \times

This leaves five potential skew lines left. Let's mark these on the diagram.

Therefore, the lines that are skew to $N Q$ are: $M L,$ $P S,$ $R S,$ $K L,$ and $L S .$

d If you were asked to name a street that is parallel to your own street, you wouldn't name your own street. The same logic applies here. We do not mention

N Q

as being parallel with, intersecting, or skew with itself.

Subchapter links

Communicate Your Answer p.125

Monitoring Progress p.126-128

Exercises p.129-130