Points, Lines, Planes, and Their Intersections | Geometry | LevelKite

Geometry starts with three objects: the point, the line, and the plane. They are undefined terms, which means geometry does not define them with simpler words. You describe them instead, and then you build every other idea from them.

You also need a starting place for reasoning. A postulate is a statement you accept without proof. This lesson uses the point-line-plane postulates, and you will use each one by its exact name. Every reason here is a definition or one of these postulates.

Describing and naming the three undefined terms

Term	What it is	How you name it
Point	One exact location with no size	One capital letter: point $A$
Line	A straight path of points with no thickness that continues without end in two opposite directions	Two of its points: line $AB$ or $\overleftrightarrow{AB}$ . Or one lowercase letter: line $\ell$
Plane	A flat surface with no thickness that continues without end in all directions	One script capital letter: plane $\mathcal{M}$ . Or three points not all on one line: plane $ABC$

A name has to pick out exactly one object. The Two Point Postulate says that through any two points there exists exactly one line. So any two points on a line name that line, in either order.

The Three Point Postulate says that through any three points not all on one line there exists exactly one plane. So three such points name that plane, in any order. If three points all lie on one line, this postulate does not apply to them, and they do not pick out one plane.

Not every straight figure is a line, and the difference is in the endpoints. A line has no endpoints. A segment is the part of a line made up of two points, called its endpoints, and all the points between them. A ray is the part of a line that starts at one endpoint and continues without end in one direction. The figure below shows line $AB$ , written $\overleftrightarrow{AB}$ ; segment $CD$ , written $\overline{CD}$ ; and ray $EF$ , written $\overrightarrow{EF}$ , which starts at $E$ and passes through $F$ .

Like every figure in this course, it is not drawn to scale. Only tick, arrow, and right-angle marks and the facts stated in the text are givens. You never read a length, an angle measure, parallel or perpendicular lines, or equal angles from a drawing. The figures in this lesson carry no marks, so all of their facts come from the text.

Line

AB

has no endpoints, segment

CD

has two, and ray

EF

has one.

Worked example

Name a point, a line, and a plane

Points $A$ , $B$ , and $C$ lie on line $\ell$ . Point $D$ does not lie on line $\ell$ . All four points lie in plane $\mathcal{M}$ . Using only these labels, name each point, then list the names of the line and the names of the plane.

Name the points first. Each point takes one capital letter as its name, so the four labels in the problem already name them.
$\text{point } A,\ \text{point } B,\ \text{point } C,\ \text{point } D$
Name the line. The problem gives it a lowercase name. By the Two Point Postulate, any two of its points also name it, in either order, so use each pair of A, B, and C both ways.
$\ell,\ \overleftrightarrow{AB},\ \overleftrightarrow{BA},\ \overleftrightarrow{AC},\ \overleftrightarrow{CA},\ \overleftrightarrow{BC},\ \overleftrightarrow{CB}$
Name the plane. The problem gives it a script name. Three points not all on one line also name it, by the Three Point Postulate. The line in the problem is the only line through any two of A, B, and C, by the Two Point Postulate, and D is given to be off it, so any two of them together with D work.
$\text{plane } \mathcal{M},\ \text{plane } ABD,\ \text{plane } ACD,\ \text{plane } BCD$
Rule out plane ABC. A, B, and C all lie on one line, so the Three Point Postulate does not apply to them, and that name does not pick out one plane.

Answer

The points are $A$ , $B$ , $C$ , and $D$ . The line is line $\ell$ , or line $AB$ , $BA$ , $AC$ , $CA$ , $BC$ , or $CB$ . The plane is plane $\mathcal{M}$ , or plane $ABD$ , $ACD$ , or $BCD$ with the letters in any order, but not plane $ABC$ .

Now you can ask how points relate to one another. Collinear points lie on one line, and coplanar points lie in one plane. Points that do not all lie on one line are noncollinear, and points that do not all lie in one plane are noncoplanar. A definition works in both directions: points that lie on one line are collinear, and collinear points lie on one line. The definition of coplanar works the same way.

Two cases are automatic. Any two points are collinear, by the Two Point Postulate. Any three noncollinear points are coplanar, by the Three Point Postulate. Neither fact goes further: three points need not be collinear, and four points need not be coplanar. For those, you need given facts, not the look of a drawing.

Key concept

Ask which points lie where

Collinear, coplanar, and intersection all ask one question: which points lie on which figures? Collinear points share a line. Coplanar points share a plane. The intersection of two or more figures is the set of all the points they have in common. Answer from given facts, definitions, and named postulates, never from how a drawing looks.

Here is how to use that idea with a figure. A plane is often pictured by a four-sided outline that shows only part of it. In the figure below, the four segments outline part of plane $ABC$ , and the plane continues past them because a plane has no edges. The figure is not drawn to scale and has no marks, so these given facts tell you where the points lie:

Points $A$ , $B$ , and $C$ do not lie on one line.
Points $D$ , $E$ , and $F$ also lie in plane $ABC$ .
Point $G$ lies on line $EF$ , and point $A$ does not.
Point $H$ does not lie in plane $ABC$ .

Line

EF

passes through

G

. Point

G

is drawn past the outline and point

H

inside it, so use the given facts, not those positions.

Worked example

Test points for collinear and coplanar

Use the figure and its given facts. Are $E$ , $F$ , and $G$ collinear? Are $A$ , $E$ , and $F$ collinear? Are $A$ , $B$ , $C$ , and $G$ coplanar? Are $A$ , $B$ , $C$ , and $H$ coplanar?

Test E, F, and G. E and F lie on line EF by its name, and G is given to lie on that line. All three lie on one line, so they are collinear by the definition of collinear.
$E,\ F,\ G \text{ lie on } \overleftrightarrow{EF}$
Test A, E, and F. By the Two Point Postulate, line EF is the only line through E and F. A is given to be off that line, so no line contains all three, and they are noncollinear.
$A \text{ is not on } \overleftrightarrow{EF}$
Test A, B, C, and G. The Plane-Line Postulate says that if two points lie in a plane, then the line containing them lies in the plane. E and F lie in plane ABC, so line EF does too, and G lies on line EF. All four points lie in plane ABC, so they are coplanar by the definition of coplanar.
$\overleftrightarrow{EF} \text{ lies in plane } ABC,\ \text{so } G \text{ does too}$
Test A, B, C, and H. A, B, and C do not lie on one line, so by the Three Point Postulate plane ABC is the only plane that contains them. H is given not to lie in that plane, so no plane contains all four, and they are noncoplanar.
$H \text{ is not in plane } ABC$

Answer

$E$ , $F$ , and $G$ are collinear. $A$ , $E$ , and $F$ are noncollinear. $A$ , $B$ , $C$ , and $G$ are coplanar. $A$ , $B$ , $C$ , and $H$ are noncoplanar.

Figures intersect when they have at least one point in common. The table below lists every possible intersection for two different lines, for a line and a plane, and for two different planes. No point appears in every row, so figures do not have to intersect. Decide what figures share from their names and the given facts, because a crossing on a page does not prove a shared point.

What an intersection can be

Figures	Possible intersections	Reason
Two different lines	No point, or exactly one point	Line Intersection Postulate: if two lines intersect, their intersection is exactly one point.
A line and a plane	No point, exactly one point, or the whole line	Plane-Line Postulate: two shared points put the whole line in the plane.
Two different planes	No point, or exactly one line	Plane Intersection Postulate: if two planes intersect, their intersection is a line.

The figure below pictures three kinds of intersection at once. Its two outlines show parts of plane $ABC$ and plane $ABE$ , like the open pages of a book. Each outline stops only in the drawing, because the planes continue without end. The figure is not drawn to scale and has no marks, so these given facts tell you where the points lie:

Points $A$ , $B$ , and $C$ do not lie on one line, and neither do $A$ , $B$ , and $E$ .
Point $D$ also lies in plane $ABC$ .
Points $F$ and $G$ also lie in plane $ABE$ .
Point $G$ does not lie in plane $ABC$ .

The left outline shows part of plane

ABC

and the right outline shows part of plane

ABE

. Line

BG

passes through

B

and

G

.

Worked example

Find three intersections

Use the figure and its given facts. Find the intersection of plane $ABC$ and plane $ABE$ . Then find the intersection of line $AB$ and line $BG$ , and the intersection of line $BG$ and plane $ABC$ .

Start with the two planes. They are different, because G lies in plane ABE but not in plane ABC. Both contain A and B by their names, so they intersect, and the Plane Intersection Postulate says their intersection is a line. That line contains A and B, and by the Two Point Postulate the only such line is line AB.
$\text{plane } ABC \text{ and plane } ABE \text{ intersect in } \overleftrightarrow{AB}$
Show that line AB and line BG are different lines. A and B lie in plane ABC, so the Plane-Line Postulate puts line AB in plane ABC. G does not lie in plane ABC, so G is not on line AB.
$G \text{ is not on } \overleftrightarrow{AB}$
Both lines contain B by their names, so they intersect. Because they are different lines, the Line Intersection Postulate says their intersection is exactly one point, and that point is B.
$\overleftrightarrow{AB} \text{ and } \overleftrightarrow{BG} \text{ intersect at } B$
Now take line BG and plane ABC. They share B. A second shared point would put all of line BG in plane ABC, by the Plane-Line Postulate, and G would then lie in plane ABC. The given facts say it does not, so B is the only shared point.
$\overleftrightarrow{BG} \text{ and plane } ABC \text{ intersect at } B$

Answer

Plane $ABC$ and plane $ABE$ intersect in line $AB$ . Line $AB$ and line $BG$ intersect at point $B$ . Line $BG$ and plane $ABC$ intersect at point $B$ .

Takeaway: you name points, lines, and planes with letters, and every question about them comes down to which points lie on which figures. Decide from given facts, definitions, and named postulates, never from the look of a drawing. Next, you will model points, lines, and planes in the real world and use these undefined terms in precise definitions.