Lesson 0.1
Points, Lines, Planes, Segments, and Rays
Geometry begins with three undefined terms — point, line, and plane — plus defined figures built from them: segments, rays, and their relationships.
Every geometric figure — from triangles to circles to three-dimensional solids — is built from points, lines, and planes. Mastering how to name and distinguish these fundamental objects is the foundation for every proof and construction in geometry.
Essential Question
How can points, lines, planes, segments, and rays be represented and named accurately?
Learning Goals
- 1Identify and describe points, lines, and planes.
- 2Name lines, segments, rays, and planes using correct geometric notation.
- 3Distinguish among a line, a line segment, and a ray.
- 4Identify collinear and coplanar points.
- 5Identify intersections of lines and planes.
- 6Recognize and name opposite rays.
- 7Interpret basic geometric diagrams accurately.
Key Vocabulary
Point
An exact location with no size, length, width, or thickness. Named with a capital letter.
Line
A straight path that extends forever in two opposite directions and has no thickness.
Plane
A flat surface that extends forever in all directions and has no thickness.
Collinear points
Points that lie on the same line.
Noncollinear points
Points that do not all lie on the same line.
Coplanar points
Points that lie in the same plane.
Line segment
A part of a line with two endpoints.
Endpoint
A point at the end of a segment or the starting point of a ray.
Ray
A part of a line that begins at one endpoint and extends forever in one direction.
Opposite rays
Two rays that share the same endpoint and extend in opposite directions to form a line.
Intersection
The point or set of points shared by two or more geometric figures.
Space
The set of all points in three dimensions.
Key Ideas
Geometry begins with three undefined terms — point, line, and plane. These are described informally rather than defined using simpler geometric terms. Every other geometric definition traces back to these three ideas.
A. Undefined Terms
- A point, a line, and a plane are the three undefined terms of geometry.
- They are described, not formally defined, because no simpler geometric terms exist to define them.
- All other geometric figures and definitions are built from these three concepts.
B. Points
- A point represents an exact location in space with no dimensions — no length, width, or thickness.
- Shown as a small filled dot; named with a single capital letter (e.g., point A).
- A point has no size; it only indicates position.
C. Lines
- A line extends forever in two opposite directions and has no thickness.
- Drawn with arrowheads at both ends to show it continues infinitely.
- Named using any two points on it (order does not matter) or a lowercase script letter.
- Valid names for a line through A and B: line AB, line BA, or line ℓ.
D. Planes
- A plane is a flat, two-dimensional surface extending forever in all directions.
- Named by a capital script letter or by three noncollinear points in the plane.
- Three noncollinear points determine exactly one plane.
- Three collinear points cannot name a plane — they lie on a line contained in infinitely many planes.
E. Collinear and Coplanar Points
- Collinear points lie on one line; coplanar points lie on one plane.
- Any two points are always collinear; any three points are always coplanar.
- Three noncollinear points determine exactly one plane.
F. Segments
- A line segment (or segment) is a part of a line with two endpoints.
- Segment AB has endpoints A and B and a measurable, finite length.
- Segment AB = segment BA — the two endpoints may be listed in either order.
- The notation AB̄ names the geometric figure; AB (no bar) names its length.
G. Rays
- A ray has one endpoint and extends forever in one direction.
- The endpoint must always be written first: ray AB starts at A and extends through B.
- Ray AB ≠ Ray BA — they start at different points and point in opposite directions.
H. Opposite Rays
- Opposite rays share the same endpoint and extend in exactly opposite directions.
- Together, two opposite rays form a complete straight line.
- If B is between A and C on a line, then ray BA and ray BC are opposite rays.
I. Intersections
- Two distinct lines intersect at exactly one point.
- Two distinct planes intersect in a line.
- A line and a plane may intersect at one point, or the line may lie entirely within the plane.
- Parallel lines or planes do not intersect — they share no points.
Worked Examples
A line contains points A, B, and C. Give two valid names for the line.
Step 1. Choose any two of the three points on the line.
Step 2. Write the two letters in either order — order does not matter for a line.
Step 3. Valid names include: line AB, line BA, line AC, line CA, line BC, line CB.
Points P, Q, R are collinear with Q between P and R. Name: (a) segment PQ, (b) ray QR, (c) the ray opposite to ray QR.
(a) A segment is named by its two endpoints in either order: segment PQ.
(b) The endpoint must come first: ray QR (starts at Q, extends through R).
(c) The opposite ray shares endpoint Q and points toward P: ray QP.
Points D, E, F lie on one line. Point G is above the line. Which points are collinear?
Step 1. Collinear means all points lie on the same line.
Step 2. D, E, and F all lie on the same line — they are collinear.
Step 3. G does not lie on that line — it is above it.
A plane contains noncollinear points J, K, L, and also point M. Give two valid names for the plane.
Step 1. A plane is named by three noncollinear points in it.
Step 2. J, K, L are noncollinear → plane JKL is valid.
Step 3. M is also in the plane → plane JKM is also valid.
Two lines cross at point T. What is the intersection of the two lines?
Two distinct lines share at most one point.
Multiple Choice Check
Circle the letter of the best answer for each question.
The diagram below shows three figures. Which row correctly identifies all three?
- ALine, segment, ray
- BRay, line, segment
- CSegment, ray, line
- DRay, segment, line
In the diagram below, points D, E, F lie on one line and point G is above the line. Which statement is true?
- AD, E, F, and G are all collinear.
- BD, E, and F are collinear; G is noncollinear with them.
- COnly D and E are collinear.
- DG is collinear with D and F only.
The diagram below shows points A, B, and C on a line with B between A and C. Which pair of rays are opposite rays?
- ARay AB and ray AC
- BRay BA and ray BC
- CRay AB and ray BC
- DRay BA and ray CA
Points R, S, T, and U are all in the same plane. Points R, S, and T are collinear. Which name correctly identifies the plane?
- APlane RST
- BPlane RSU
- CPlane R
- DPlane RS
Two distinct planes intersect. What is always true about their intersection?
- AThey intersect at exactly one point.
- BThey intersect along a line.
- CThey intersect along a segment.
- DThey intersect at two points.
Guided Practice
Work through each question. Expand the solution to check your reasoning.
A line contains points M, N, and P. List all valid two-letter names for this line.
Points A, B, C are collinear with B between A and C. Name the two opposite rays with endpoint B.
Explain why ray AB and ray BA are not the same ray.
A plane contains noncollinear points J, K, L, and M. Is 'plane JKL' valid? Is 'plane JKM' valid?
Segment AB has a length of 8 cm. What is the difference between the notation for the segment and the notation for its length?
Common Mistakes
Naming a ray with its endpoint second (e.g., writing "ray BA" when you mean the ray starting at A).
The endpoint must always be written first. Ray AB starts at A; ray BA starts at B. These are different rays.
Thinking segment AB and ray AB represent the same figure.
A segment has two endpoints and a finite length. A ray has one endpoint and extends forever. They are different figures.
Using only one point to name a line.
A line is named with two points on it (e.g., line AB) or a lowercase script letter. A single point does not identify a unique line.
Naming a plane with three collinear points.
Three collinear points lie on one line, which is contained in infinitely many planes. Use three noncollinear points to identify a unique plane.
Writing "segment AB = 8" (equating a geometric figure to a number).
The bar notation names the figure; without the bar, the letters represent the length. Write "AB = 8" for the length.
Assuming figures stop where the diagram stops.
Arrowheads indicate that lines and rays continue beyond the visible drawing. A line with two arrowheads extends forever in both directions.
Before You Finish
Lesson Summary
- Point, line, and plane are the three basic undefined terms of geometry.
- A segment has two endpoints and a measurable length.
- A ray has one endpoint and extends forever in one direction.
- A line has no endpoints and extends forever in both directions.
- Collinear points lie on one line; coplanar points lie on one plane.
- The endpoint of a ray must always be written first.
- Opposite rays share one endpoint and together form a straight line.
- Intersections describe the points shared by two or more geometric figures.
Check Your Understanding
1. How is a ray different from a segment?
2. Why must the endpoint of a ray be named first?
3. What is the intersection of two distinct planes?