Exercise 12 Page 233

a Parallel segments lie in the same plane but do not intersect. On the diagram below, the faces containing segment

\overline{M Q}

are highlighted.

There are two segments parallel to $\overline{M Q},$ segments $\overline{O S}$ and $\overline{N R} .$

b Let's look at faces of the prism and diagonal planes separately.

Faces of the Prism

All faces of this prism intersect plane $S R N .$ One of these is shaded red on the diagram below.

The table below contains all faces of the prism other than $S R N .$ Since three noncollinear points determine a plane, some planes might be described using different points. The table also contains these alternative descriptions.

Plane	Alternative Description
$S Q O$	$S Q M,$ $S M O,$ $Q M O$
$S Q R$
$R Q M$	$R Q N,$ $R M N,$ $Q M N$
$O M N$

Diagonal Planes

There are also diagonal planes that intersect plane $S R N .$ One of these is shaded green on the diagram below.

All of these diagonal planes are determined by two points on plane $S R N$ and one point outside this plane. The table below contains all possibilities that are not considered already among the faces of the prism.

Points on plane $S R N$	Intersecting Plane(s)
$R,$ $O$	$R O Q,$ $R O M$
$R,$ $S$	$R S M$
$R,$ $N$	face
$S,$ $O$	face
$S,$ $N$	$S N Q,$ $S N M$
$O,$ $N$	$O N Q$

Summary

There are four faces and six diagonal planes, so altogether there are ten planes intersecting plane $S R N .$

c We are looking for segments that are not coplanar with segment

\overline{O N} .

On the diagram below the two faces of the prism containing segment

\overline{O N}

are shaded green.

We need a segment not in these planes and not intersecting $\overline{O N} .$ There are three segments like this, all going through vertex $Q .$ The three segments skew to $\overline{O N}$ are $\overline{Q R},$ $\overline{Q S},$ and $\overline{Q M} .$

Subchapter links

Exercises p.232-233