Parallel, Perpendicular, and Skew Lines and Parallel Planes | Geometry | LevelKite

You already know how to find where lines and planes intersect. In this lesson, you will use that skill to name how two lines, or two planes, are related. Each name has an exact definition, and one word appears in several of them: two lines are coplanar when one plane contains both of them. Throughout this lesson, "two lines" means two different lines and "two planes" means two different planes.

Start with two lines in one plane. Parallel lines are two coplanar lines that do not intersect. A line goes on forever, so this includes places beyond the edge of a drawing. Matching arrow marks on two lines state that they are parallel. You write $\overleftrightarrow{AB} \parallel \overleftrightarrow{CD}$ , where $\overleftrightarrow{AB}$ means line $AB$ .

Perpendicular lines are two lines that intersect to form a right angle, which is an angle that measures exactly $90^\circ$ . A small square at the intersection states a right angle. You write $\overleftrightarrow{AB} \perp \overleftrightarrow{AE}$ .

In the figure below, the arrows and the square state these two facts. The way the lines look does not.

The arrows state that line

AB

is parallel to line

CD

. The square states a right angle at

A

. No mark relates line

AE

and line

FG

.

Worked example

Reading marks in one plane

Given: all four lines in the figure lie in one plane. The figure is not drawn to scale, so use only that stated fact and the marks. Classify each pair: line $AB$ and line $CD$ ; line $AB$ and line $AE$ ; line $AE$ and line $FG$ .

Line AB and line CD carry matching arrow marks, and those marks state that the lines are parallel. The reason is the marks, not the even gap you may see between the lines.
$\overleftrightarrow{AB} \parallel \overleftrightarrow{CD}$
Line AB and line AE both pass through A, so they intersect there. The square mark at A states a right angle, so by the definition of perpendicular lines, they are perpendicular.
$\overleftrightarrow{AB} \perp \overleftrightarrow{AE}$
Line AE and line FG may look parallel, but no arrow mark or stated fact says so. How they look is not a reason, so you cannot tell whether they intersect somewhere off the page. This pair stays unclassified.

Answer

Line $AB$ is parallel to line $CD$ , line $AB$ is perpendicular to line $AE$ , and the givens do not classify line $AE$ and line $FG$ .

Now leave the flat page. In space, two lines can miss each other without being parallel. Picture a straight, level road on a bridge above another straight, level road that runs in a different direction. On a flat map the roads seem to cross. Modeled as lines, they do not meet, and no single plane contains both.

Skew lines are two lines that are not coplanar. They cannot intersect, because by a theorem, two intersecting lines lie in exactly one plane. They cannot be parallel either, because the definition of parallel lines requires one plane that contains both lines.

Planes have a matching word. Parallel planes are two planes that do not intersect, so they share no point. If two planes share a point, they are not parallel, and by the postulate on intersecting planes, they meet in a line.

Key concept

Two questions sort any two lines

For two different lines, ask these questions in order.

1. Do they intersect? If yes, they are perpendicular when they form a right angle, and otherwise they are simply intersecting lines.

2. If they do not intersect, are they coplanar? If yes, they are parallel. If no, they are skew.

For two different planes, ask only whether they intersect. If they do not, they are parallel.

Answer only from marks and stated facts. A figure is not drawn to scale, so its look never answers a question. If the givens do not answer it, the pair stays unclassified.

The figure below is a flat sketch that stands for a see-through rectangular box. The front face is $ABCD$ and the back face is $EFGH$ , with $E$ behind $A$ , $F$ behind $B$ , $G$ behind $C$ and $H$ behind $D$ . That makes $ABFE$ the top face, $DCGH$ the bottom face, $ADHE$ the left face and $BCGF$ the right face. Lines $AB$ and $EH$ are drawn past the box.

These facts are stated, so they are givens:

Each of the six faces is a rectangle. So each face lies in one plane, its neighboring sides form right angles, and its opposite sides are parallel.
Opposite faces lie in parallel planes.

The sketch is not drawn to scale, so read nothing else from it. A side of a face is a segment, and when you classify sides, you classify the lines that contain them. You can name a plane by three of its points that are not on one line. For example, plane $ABC$ contains the front face.

A flat box sketch, not to scale. The arrows state that line

AB

and line

DC

are parallel, and the square states a right angle at

A

. All other facts come from the text.

Worked example

Using stated facts about the box

Given: the stated box facts. The sketch is not drawn to scale, so do not use its look. Classify each pair: plane $ABC$ and plane $EFG$ ; plane $ABC$ and plane $ABF$ ; line $AB$ and line $EF$ ; line $AB$ and line $BF$ .

Plane ABC contains the front face and plane EFG contains the back face. These are opposite faces, so the stated fact says that their planes are parallel.
$\text{plane } ABC \parallel \text{plane } EFG$
Plane ABC contains the front face and plane ABF contains the top face. Both planes contain A and B, so they intersect. By the definition of parallel planes, they are not parallel, and they meet in line AB.
$\text{plane } ABC \cap \text{plane } ABF = \overleftrightarrow{AB}$
Line AB and line EF contain opposite sides of rectangle ABFE. The stated fact about faces says that opposite sides are parallel.
$\overleftrightarrow{AB} \parallel \overleftrightarrow{EF}$
Line AB and line BF both pass through B, so they intersect. They contain neighboring sides of rectangle ABFE, so the stated fact gives a right angle at B, even though the sketch does not draw that corner square. By the definition of perpendicular lines, they are perpendicular.
$\overleftrightarrow{AB} \perp \overleftrightarrow{BF}$

Answer

Plane $ABC$ is parallel to plane $EFG$ , plane $ABC$ is not parallel to plane $ABF$ , line $AB$ is parallel to line $EF$ , and line $AB$ is perpendicular to line $BF$ .

Worked example

Showing that two lines are skew

Given: the stated box facts. To show: line $AB$ and line $EH$ are skew. The two lines cross in the sketch, but that crossing is neither a mark nor a stated fact, and the sketch is not drawn to scale.

First question: do the lines intersect? By the postulate that a line through two points of a plane lies in that plane, line AB lies in plane ABC and line EH lies in plane EFG. Those planes contain opposite faces, so they are parallel and share no point. So the lines do not intersect.
Second question: are the lines coplanar? Suppose one plane contains both lines. Then that plane contains A, B, E and H.
Sides AB and AE of rectangle ABFE form a right angle, so A, B and E are not on one line. By the postulate that exactly one plane passes through three points not on one line, the supposed plane must be plane ABF, the plane of the top face.
H is a corner of the bottom face. The top and bottom faces are opposite, so their planes are parallel and share no point, by the definition of parallel planes. So H is not in plane ABF.
$H \notin \text{plane } ABF$
The supposed plane must contain H and must be plane ABF, which is impossible. So no plane contains both lines. They are not coplanar, so by the definition of skew lines, they are skew.
$\overleftrightarrow{AB} \text{ and } \overleftrightarrow{EH} \text{ are skew}$

Answer

Line $AB$ and line $EH$ are skew, so they do not intersect, even though they cross in the sketch.

Here is the central idea. For two different lines, ask whether they intersect and then whether they are coplanar: lines that intersect at a right angle are perpendicular, coplanar lines that do not intersect are parallel, and lines that are not coplanar are skew. Two different planes that do not intersect are parallel. Base each answer on marks and stated facts, never on the look of a figure, because a figure is not drawn to scale. Next, you will meet transversals and classify the angle pairs they form.