What is the difference between AB and AB?

AB names the segment itself, a geometric figure made of points. AB with no bar means the length or distance between A and B, which is a number. So you write AB = 12 for a measurement and AB CD for two segments of equal length. Mixing them up is the single most common notation error in Unit 1.

Why are point, line, and plane called undefined terms?

Every definition uses other words, so a system of definitions has to start somewhere or it becomes circular. Geometry starts by accepting point, line, and plane as intuitive, undefined ideas that we describe rather than define. Every other term — segment, ray, angle, circle, polygon — is then built from them with a real definition.

Can two planes be skew?

No. Skew applies only to lines. Two distinct planes are either parallel (no points in common) or intersecting in exactly one line. Because planes are infinite and flat, there is no way for them to avoid each other while also failing to be parallel.

How do I tell parallel lines from skew lines in a picture of a box?

Check coplanarity, not just whether they meet. If you can name a single face — or any flat slice — that contains both lines, they are coplanar, so they are parallel or intersecting, never skew. If no flat surface contains both and they never touch, they are skew. A quick method is to eliminate every edge that shares a face with your reference edge; what remains is skew to it.

GEOM-1.1

Points, Lines & Planes

Learn to name points, lines, planes, segments, and rays, and apply the postulates for collinear points, coplanar points, intersections, and skew lines.

What you'll do in this lesson

A voice-first session with the Crimsora tutor on Points, Lines & Planes, then targeted practice and FRQs — with the tutor adapting to where you get stuck.

Start Points, Lines & Planes free

What this lesson covers

Every proof, construction, and formula you meet in Geometry rests on three ideas that are never formally defined: point, line, and plane. Mathematicians accept them as starting places, describe them carefully, and then build everything else on top. That may sound like a technicality, but it changes how you read a diagram: a dot on paper is not really a point, and a shaded parallelogram is not really a plane — each is a picture of something that extends beyond the page or has no size at all.

In this lesson you will learn the correct symbols and names for points, lines, planes, segments, and rays, and then use the basic postulates that tell you when points determine a line or a plane, how two lines or two planes meet, and what makes two lines skew rather than parallel. Getting the notation exact now saves you from a lot of confusion when you start measuring and proving in the lessons ahead.

The Three Undefined Terms and How to Name Them

A point has position but no size — no length, width, or thickness. We draw it as a dot and name it with a single capital letter, like point

A

.

A line is a straight arrangement of points that extends forever in two directions. It has infinite length but no width. Name a line either with a lowercase script letter, like line

m

, or with any two points on it:

\overleftrightarrow{AB}

. Order does not matter, so

\overleftrightarrow{AB}

and

\overleftrightarrow{BA}

are the same line.

A plane is a flat surface that extends forever in all directions, with no thickness. Name it with a capital script letter, like plane

\mathcal{P}

, or with any three points in it that do not lie on the same line: plane

ABC

Figure	Symbol	How to name it
Point	$A$	one capital letter
Line	$\overleftrightarrow{AB}$	two points, any order
Plane	plane $ABC$	three noncollinear points
Segment	$\overline{AB}$	two endpoints, any order
Ray	$\overrightarrow{AB}$	endpoint first, then any other point

A common early mistake is dropping the bar or arrow. Writing

AB

with no symbol above it means the distance between

A

and

B

— a number — while

\overline{AB}

means the figure itself. So "

\overline{AB} = 7

" is sloppy; "

AB = 7

" is correct, and "

\overline{AB} \cong \overline{CD}

" is how you say two segments are the same size. Another mistake is naming a plane with three collinear points, which does not pin down a single plane at all.

Segments, Rays, and Opposite Rays

Both segments and rays are pieces of a line, so both are defined using points on that line.

A segment

\overline{AB}

consists of endpoints

A

and

B

together with every point of

\overleftrightarrow{AB}

between them. Because both ends are endpoints, order does not matter:

\overline{AB}

and

\overline{BA}

name the same segment.

A ray

\overrightarrow{AB}

starts at endpoint

A

, passes through

B

, and continues forever past

B

. Here order matters enormously.

\overrightarrow{AB}

and

\overrightarrow{BA}

are different rays — they start at different points and travel in opposite directions. When you name a ray, the endpoint is always written first, and the arrow symbol always points right regardless of which way the ray goes in the picture.

Two rays are opposite rays when they share the same endpoint and their union is a whole line. That requires three collinear points: if

B

is between

A

and

C

on a line, then

\overrightarrow{BA}

and

\overrightarrow{BC}

are opposite rays. Students often claim

\overrightarrow{AB}

and

\overrightarrow{BA}

are opposite rays. They are not — they overlap on all of

\overline{AB}

and have different endpoints. Opposite rays must share one endpoint and overlap nowhere else.

One more subtlety: a ray can be named using any point on it other than the endpoint. If

A

B

, and

C

are collinear in that order, then

\overrightarrow{AB}

and

\overrightarrow{AC}

are the exact same ray, because both start at

A

and head the same direction. Recognizing when two different names describe one figure shows up constantly in homework problems that ask "how many different rays appear in this diagram?"

Collinear, Coplanar, and the Determining Postulates

Points are collinear if one line passes through all of them, and coplanar if one plane contains all of them. Any two points are automatically collinear — you can always draw a line through them. Any three points are automatically coplanar. The interesting questions start at three points for collinearity and four points for coplanarity.

These postulates are the rules you cite:

Through any two points there is exactly one line. Through any three noncollinear points there is exactly one plane. A line contains at least two points; a plane contains at least three noncollinear points. If two points lie in a plane, then the entire line through them lies in that plane.

That last one is the workhorse. It explains why a plane is flat: you cannot have two of its points connected by a line that pokes out of it.

Why "noncollinear" in the plane postulate? Picture three points on one straight line and imagine planes containing them — you can spin infinitely many planes around that line, like pages rotating around the spine of a book. The three points do not determine one plane. Move a point off the line and the spinning stops; exactly one plane fits.

This is also the reason a three-legged stool never wobbles while a four-legged one might. Three leg tips are noncollinear, so they always determine exactly one plane. A fourth tip may or may not lie in it.

When a problem asks you to name three collinear points from a diagram, trace along a single drawn line. When it asks for four noncoplanar points, look for a point that sticks out of the shaded region — typically the apex of a pyramid or a vertex on the opposite face of a box.

Intersections and Skew Lines

An intersection is the set of points that two figures have in common. Two postulates cover the standard cases: if two distinct lines intersect, they intersect in exactly one point; if two distinct planes intersect, they intersect in exactly one line.

That second fact surprises people. Two walls of a room do not meet at a point — they meet along the whole vertical edge where they join. Planes are infinite and flat, so once they share two points they must share the entire line through those points.

Figures	Possible intersections
Two lines	one point, or none (parallel or skew), or the same line
Line and plane	one point, or none, or the whole line lies in the plane
Two planes	one line, or none (parallel), or the same plane

Skew lines are lines that do not intersect and are not coplanar. This is the pair of conditions students most often mishandle. Parallel lines also never intersect, but parallel lines are coplanar; skew lines are not. If you can find a single flat surface containing both lines, they are not skew.

A rectangular box is the best model. Take the front-bottom edge and the right-vertical edge on the back face: they never meet, and no flat sheet can contain both, so they are skew. The front-bottom edge and the back-bottom edge, however, lie in the bottom face together, so those are parallel.

Also note that only lines can be skew. Two planes are either parallel or intersecting — never skew. And a line and a plane are never called skew either. When a question asks for "all edges skew to

\overline{AB}

," first eliminate every edge that touches

\overline{AB}

, then eliminate every edge sharing a face with it; whatever remains is skew.

Reading Three-Dimensional Diagrams

Most errors in this topic are diagram-reading errors, not concept errors. Textbook pictures of solids are flat drawings of three-dimensional objects, and your eye will lie to you.

First, dashed lines represent hidden edges — parts of the figure behind the front faces. They are still real edges of the solid, so a dashed edge counts when you list parallel, intersecting, or skew lines.

Second, two segments that appear to cross in a drawing may not actually intersect. In a picture of a box, the front-top edge and a back-vertical edge can visually overlap on the page while being far apart in space. Ask whether they share an actual labeled point, not whether the ink crosses.

Third, planes drawn as parallelograms extend beyond their borders. If a question asks whether point

P

lies in plane

\mathcal{M}

, the drawn boundary is irrelevant; what matters is whether

P

is on the flat surface the parallelogram represents, extended infinitely.

Fourth, in a plane figure you may assume points that appear on a drawn line are collinear and that the drawn arrangement is accurate, but you may not assume anything about measurements or right angles from appearance alone. Lesson 1.2 and 1.3 make that distinction sharper when segment and angle measures enter the picture.

A practical routine: label everything. Before answering, write the letters of each face, trace each edge with a finger, and say out loud which plane contains it. Students who list "the planes containing edge

\overline{BC}

" before hunting for skew lines rarely miss one, because every edge sharing a plane with

\overline{BC}

is automatically disqualified from being skew to it.

Key terms

Point.: A location with no length, width, or thickness, drawn as a dot and named with a single capital letter.
Line.: An infinite set of points extending forever in two directions with no thickness; named $\overleftrightarrow{AB}$ or by a lowercase letter.
Plane.: A flat surface extending infinitely in all directions with no thickness; named by a capital script letter or by three noncollinear points.
Segment.: Two endpoints $A$ and $B$ plus all points between them on $\overleftrightarrow{AB}$ , written $\overline{AB}$ .
Ray.: A part of a line starting at an endpoint and extending forever through another point; $\overrightarrow{AB}$ begins at $A$ .
Opposite rays.: Two rays with a common endpoint whose union is an entire line, such as $\overrightarrow{BA}$ and $\overrightarrow{BC}$ when $B$ is between $A$ and $C$ .
Collinear / Coplanar.: Points are collinear if one line contains them all; coplanar if one plane contains them all.
Skew lines.: Lines that neither intersect nor lie in the same plane; they exist only in three dimensions.

Worked example

A rectangular box has top face

ABCD

and bottom face

EFGH

, with

A

above

E

B

above

F

C

above

G

, and

D

above

H

. (a) Name the intersection of plane

ABCD

and plane

ABFE

. (b) Name two edges parallel to

\overline{AB}

. (c) Name all edges skew to

\overline{AB}

. (d) Are points

A

B

G

, and

H

coplanar?

(a) Two distinct planes intersect in exactly one line. Plane

ABCD

is the top; plane

ABFE

is the front. The points they share are

A

and

B

, so the intersection is

\overleftrightarrow{AB}

— the front-top edge.

(b) Parallel edges never meet and share a plane with

\overline{AB}

. On the top face,

\overline{DC}

is parallel to

\overline{AB}

. On the bottom face,

\overline{EF}

lies directly below

\overline{AB}

and is parallel to it. (

\overline{HG}

is a third one.)

(c) Work by elimination. Edges that touch

\overline{AB}

at an endpoint cannot be skew:

\overline{AD}

\overline{AE}

\overline{BC}

\overline{BF}

. Edges parallel to it cannot be skew:

\overline{DC}

\overline{EF}

\overline{HG}

. That leaves

\overline{DH}

\overline{CG}

\overline{EH}

, and

\overline{FG}

. Check each:

\overline{DH}

and

\overline{CG}

are vertical edges on the back face — they miss

\overline{AB}

and no flat surface holds both, so they are skew.

\overline{EH}

and

\overline{FG}

are bottom edges running front-to-back; they never meet

\overline{AB}

and are not coplanar with it, so they are skew too. Answer:

\overline{DH}

\overline{CG}

\overline{EH}

\overline{FG}

.

(d) Yes.

\overline{AB}

is the front-top edge and

\overline{HG}

is the back-bottom edge. A single slanted plane cuts diagonally through the box containing all four points, so

A

B

G

H

are coplanar even though they are not a face.

Practice questions

Which statement about

\overrightarrow{PQ}

and

\overrightarrow{QP}

is true?

They are the same ray.
They are opposite rays.
They are two different rays whose union is $\overleftrightarrow{PQ}$ .
They do not overlap at any point.

Answer: They are two different rays whose union is $\overleftrightarrow{PQ}$ .

\overrightarrow{PQ}

starts at

P

and heads through

Q

forever;

\overrightarrow{QP}

starts at

Q

and heads through

P

forever. Different endpoints means they are not the same ray, and opposite rays must share an endpoint, so they are not opposite. They overlap along all of

\overline{PQ}

, and together they cover every point of the line, so their union is

\overleftrightarrow{PQ}

Points

R

S

, and

T

are collinear. Explain why these three points do not determine exactly one plane, and state what change would make them determine one.

Answer: Infinitely many planes contain a single line, so three collinear points lie in infinitely many planes; moving one point off the line makes the three points noncollinear, and then exactly one plane contains them.

All three points lie on one line

\overleftrightarrow{RS}

. Imagine that line as the spine of a book: every page rotated around it is a plane containing

R

S

, and

T

. Since infinitely many such planes exist, the points do not determine a unique one. The postulate requires three noncollinear points, because a third point off the line locks the rotation into a single position — this is exactly why a three-legged stool sits flat.

In a triangular pyramid with base

\triangle BCD

and apex

A

, is

\overline{AB}

skew to

\overline{CD}

? Justify your answer using the definition of skew.

Answer: Yes. $\overline{AB}$ and $\overline{CD}$ do not intersect and are not coplanar, so they are skew.

Skew requires two conditions. First, no intersection:

\overline{AB}

has endpoints

A

and

B

, and

\overline{CD}

has endpoints

C

and

D

— they share no point, and the segments pass by each other in space. Second, not coplanar: any plane containing

\overline{AB}

and

\overline{CD}

would contain all four vertices

A

B

C

D

, but a pyramid's apex is by definition not in the plane of its base. Since both conditions hold, the edges are skew. Note that checking only 'they never meet' is not enough — parallel lines never meet either.

FAQ

What is the difference between $\overline{AB}$ and $AB$ ?: $\overline{AB}$ names the segment itself, a geometric figure made of points. $AB$ with no bar means the length or distance between $A$ and $B$ , which is a number. So you write $AB = 12$ for a measurement and $\overline{AB} \cong \overline{CD}$ for two segments of equal length. Mixing them up is the single most common notation error in Unit 1.
Why are point, line, and plane called undefined terms?: Every definition uses other words, so a system of definitions has to start somewhere or it becomes circular. Geometry starts by accepting point, line, and plane as intuitive, undefined ideas that we describe rather than define. Every other term — segment, ray, angle, circle, polygon — is then built from them with a real definition.
Can two planes be skew?: No. Skew applies only to lines. Two distinct planes are either parallel (no points in common) or intersecting in exactly one line. Because planes are infinite and flat, there is no way for them to avoid each other while also failing to be parallel.
How do I tell parallel lines from skew lines in a picture of a box?: Check coplanarity, not just whether they meet. If you can name a single face — or any flat slice — that contains both lines, they are coplanar, so they are parallel or intersecting, never skew. If no flat surface contains both and they never touch, they are skew. A quick method is to eliminate every edge that shares a face with your reference edge; what remains is skew to it.

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