Chapter 2 · Reasoning and Proof

Postulates and Always, Sometimes, Never

Learn the postulates about points, lines, and planes, then use them to decide whether a statement is always, sometimes, or never true.

16 min read

A proof shows that a statement must be true by giving a reason for each step. Those reasons have to start somewhere, so geometry begins with postulates. A postulate is a statement you accept as true without proof. The point-line-plane postulates are the starting facts about how points, lines, and planes fit together. You do not prove them. You use them as reasons.

A few terms will help you read these postulates. Points are collinear when one line contains all of them, and noncollinear when no line contains all of them. Two figures intersect when they share at least one point, and their intersection is the set of all points they share. Each postulate holds only under its stated conditions, so words such as if, different, and noncollinear matter.

The point-line-plane postulates

Postulate	What it means
Through any two different points, there is exactly one line.	Two points fix a line. One line passes through them, and no second line does.
Through any three noncollinear points, there is exactly one plane.	Three points not on one line fix one plane. That is why a three-legged stool does not rock.
A line contains at least two points.	Every line has at least two points you can use to name it.
A plane contains at least three noncollinear points.	A plane always has points that do not all fit on one line.
If two different points lie in a plane, then the whole line through them lies in that plane.	A line with two points in a plane cannot leave that plane.
If two different lines intersect, then their intersection is exactly one point.	Different lines that meet share one point, never two or more.
If two different planes intersect, then their intersection is a line.	Different planes that meet share a whole line, like the edge where two walls meet.

In the figure below, you are given that points $A$ , $B$ , and $C$ are noncollinear. That fact comes from this sentence, not from the drawing. The figure draws line $AB$ , segment $AC$ , and ray $BC$ , which starts at $B$ and passes through $C$ . A segment is two points on a line, called endpoints, and all the points between them. A ray starts at one endpoint and goes on without end along a line in one direction. So the drawing shows only part of the line through $A$ and $C$ and part of the line through $B$ and $C$ .

Line

AB

, segment

AC

, and ray

BC

. The text, not the picture, gives that

A

B

, and

C

are noncollinear.

Worked example

Identify and use two postulates

Given: points $A$ , $B$ , and $C$ in the figure are noncollinear. Find (a) how many planes contain all three points and (b) how many different lines pass through at least two of the points. Name the postulate you use for each answer.

For (a), find the postulate about three points and a plane. The points are given as noncollinear, so its condition holds. The point-line-plane postulate that through any three noncollinear points there is exactly one plane gives one plane and no other.
For (b), list every pair of points, because the postulate about lines speaks of two different points at a time.
$A \text{ and } B,\quad A \text{ and } C,\quad B \text{ and } C$
Apply the point-line-plane postulate that through any two different points there is exactly one line. Each pair gives exactly one line. The figure draws only segment AC and ray BC, but the postulate gives the whole line through each of those pairs.
$\text{line } AB,\quad \text{line } AC,\quad \text{line } BC$
Check that the three lines are different. If two of them were the same line, that one line would contain A, B, and C. That contradicts the given fact that the points are noncollinear, so the lines are all different.

Answer

Exactly one plane contains all three points, by the point-line-plane postulate about three noncollinear points. Exactly three different lines pass through at least two of them, by the point-line-plane postulate about two different points: line $AB$ , line $AC$ , and line $BC$ .

You can also use postulates to judge a general statement. A case is one situation that fits the statement, such as one particular pair of lines. A statement is:

always true if it holds in every case,
sometimes true if it holds in at least one case and fails in at least one case,
never true if it fails in every case.

Earlier you showed that a conjecture, a statement you think may be true, is false by finding a counterexample, one case where it fails. Here, a counterexample shows only that a statement is not always true. It could still be sometimes true or never true.

Key concept

What each answer requires

Always and never each need a reason that covers every case, such as a postulate or a definition. Sometimes needs two cases: one where the statement is true and one where it is false. A few examples cannot settle always or never, because a true case rules out only never, and a false case rules out only always.

Worked example

An always statement

Decide whether this statement is always, sometimes, or never true: Any two different points are collinear.

Let D and E be any two different points. You assume nothing else about them, so a reason about D and E covers every case.
The point-line-plane postulate that through any two different points there is exactly one line gives a line that contains both D and E.
A definition works in both directions. Collinear points are points that one line contains, and this line contains D and E, so by the definition of collinear, D and E are collinear.

Answer

The statement is always true, because one line passes through any two different points.

Worked example

A never statement

Decide whether this statement is always, sometimes, or never true: Two different planes intersect in exactly one point.

Split all pairs of different planes into two cases: the planes do not intersect, or they do intersect. Together these cases cover every pair.
Case 1: the planes do not intersect. Then they share no point, so they do not intersect in exactly one point. The statement fails.
Case 2: the planes intersect. The point-line-plane postulate that if two different planes intersect, then their intersection is a line tells you that they share a line.
The point-line-plane postulate that a line contains at least two points tells you that this shared line has at least two points. So the intersection is not exactly one point, and the statement fails here too.

Answer

The statement is never true, because two different planes share either no point or a whole line with at least two points.

A sometimes answer needs a true case and a false case, and the figure below supplies both. You are given that all three lines lie in one plane, that point $E$ lies on line $AB$ , and that line $EF$ is different from line $AB$ . The arrows on lines $AB$ and $CD$ are a mark, so they give that these lines are parallel. Parallel lines are lines in one plane that do not intersect.

The arrows give that line

AB

is parallel to line

CD

. The text gives that

E

lies on line

AB

and that line

EF

is a different line.

Worked example

A sometimes statement

Decide whether this statement is always, sometimes, or never true: Two different lines in a plane intersect in exactly one point. Use the lines in the figure and the facts stated with it.

Test lines AB and EF. They are different lines and both contain E, so they intersect. The point-line-plane postulate that if two different lines intersect, then their intersection is exactly one point tells you that they share only E. The statement is true for this pair.
Test lines AB and CD. The arrows give that they are parallel, and by the definition of parallel lines, they do not intersect. They share no point, so the statement is false for this pair.
You now have one true case and one false case, which is exactly what a sometimes answer requires.

Answer

The statement is sometimes true: it holds for lines $AB$ and $EF$ but fails for lines $AB$ and $CD$ .

Your turn: decide each statement first, then read across to check your answer and reason.

Statement	Answer	Reason, using the point-line-plane postulates
All the points of a plane lie on one line.	Never	A plane contains at least three noncollinear points, and no line contains all of those points.
Two different points in a plane lie on a line in that plane.	Always	Exactly one line passes through the two points, and that whole line lies in the plane because both points do.
Three different points lie in exactly one plane.	Sometimes	True if noncollinear. False if collinear: many planes contain their line, like the pages of a book around its spine.

A postulate is a starting fact you accept without proof, and it holds only under its stated conditions. To judge a statement, give a reason that covers every case for always or never, and give one true case and one false case for sometimes. Next, you will write a paragraph proof, a proof in sentences that gives a reason for each step.

Up next

Writing a Paragraph Proof

14 min read

Continue