DWG NO. 01.1 — Lesson 1 of 5

Points, Lines, and Planes

Unit 1: Foundations of Geometry · ~15–30 min

Objective

Name points, lines, and planes using correct notation, and determine when a set of points is collinear or coplanar.

The three undefined terms

Every idea in geometry is eventually built out of three terms so basic that we don't define them — we just describe them and agree on notation.

Term	What it is	Notation
Point	An exact location. It has no size — no length, width, or thickness.	A single capital letter: point A
Line	A straight, one–dimensional path that extends infinitely in both directions.	Two points on it: line AB, or a lowercase script letter: line ℓ
Plane	A flat, two–dimensional surface that extends infinitely in every direction.	Three non–collinear points: plane ABC, or a single capital script letter: plane M

Collinear and coplanar points

Collinear points lie on the same line. Coplanar points lie on the same plane. Any two points are always collinear, and any three points are always coplanar — it's only once you have four or more points that "coplanar" becomes a real question.

Plane M contains points A, B, and C. Line ℓ passes through A and B. Point P is not in plane M.

Postulates about points, lines, and planes

Through any two points there is exactly one line.
Through any three noncollinear points there is exactly one plane.
If two points lie in a plane, then the line containing those points lies in that plane.
If two distinct lines intersect, they intersect in exactly one point.
If two distinct planes intersect, they intersect in exactly one line.

Worked Example 1 · Naming from a diagram

ProblemUse the diagram above. Name the line shown, and identify a set of three coplanar points that are not collinear.

1.The only labeled line is the one through A and B, so it is line AB (or line ℓ).

2.A, B, and C all lie in plane M, but C is not on line ℓ — so A, B, C are coplanar without being collinear.

3.Point P is drawn off the plane (shown by the dashed leader), so P is not coplanar with A, B, and C.

Line ℓ = line AB • A, B, C are coplanar, not collinear • P is not in plane M

Worked Example 2 · Always, sometimes, or never

ProblemDecide whether each statement is always, sometimes, or never true: (a) Two points are collinear. (b) Three points are coplanar. (c) A line lies in a given plane.

1.(a) Two points always determine exactly one line, so two points are always collinear.

2.(b) Three points always determine at least one plane (even three collinear points lie in infinitely many planes), so three points are always coplanar.

3.(c) A line lies in a plane only when at least two of its points lie in that plane — otherwise it just crosses the plane at one point, or misses it. So this is sometimes true.

(a) Always • (b) Always • (c) Sometimes

Guided practice

1.Give two other ways to write “line ℓ” if it passes through points A and B.
Show answer
Line AB or line BA — order doesn't matter for naming a line, since a line has no starting point.
2.Points X, Y, and Z are collinear. Is there more than one plane that contains all three? Explain.
Show answer
Yes. Three collinear points determine a line, not a unique plane — infinitely many planes can be "fanned" around that one line.
3.Plane R and plane S intersect. What must their intersection be, and why?
Show answer
A line. Two distinct planes always intersect in exactly one line — they can't meet at just a single point or overlap completely (that would make them the same plane).
4.Name three noncollinear points you can find in the room you're sitting in right now.
Show answer
Answers will vary — any three points that aren't in a straight row work, e.g. three different corners of the ceiling.
5.If two points lie in plane K, what else must lie in plane K?
Show answer
The entire line through those two points — that's exactly what the third postulate above guarantees.

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