Geometry Foundations: Points, Lines, and Planes as Building Blocks

The Three Undefined Terms

Geometry begins with three basic ideas that are treated as undefined terms: point, line, and plane. They are not defined using simpler geometric words; instead, we understand them through descriptions, diagrams, and how they behave in relationships.

Point (visual: a dot)

A point represents an exact location. In diagrams it is drawn as a small dot and labeled with a capital letter.

• A

• What it represents: a location only.
• What it does not represent: it does not have length, width, thickness, or area (no size).
• How it is named: usually by one capital letter, like A.

Line (visual: a straight path with arrows)

A line represents a perfectly straight path that extends forever in two directions. In diagrams, arrowheads show that it continues without end.

<----•----•---->  (line through A and B)

• What it represents: straightness and direction, extending infinitely.
• What it does not represent: it does not have thickness, and it does not have endpoints.
• How it is named: by two points on it (e.g., line AB or \overleftrightarrow{AB}) or by a lowercase script letter (e.g., line l).

Plane (visual: a slanted parallelogram)

A plane represents a flat surface that extends forever in all directions within that surface. In diagrams it is often drawn as a slanted parallelogram to suggest a flat sheet.

   ____________
  /          /|
 /__________/ |
 |          | /
 |__________|/   (plane P)

• What it represents: a flat surface with no thickness, extending infinitely.
• What it does not represent: it is not a finite sheet with edges; the drawn parallelogram is only a picture.
• How it is named: by a capital letter (plane P) or by three non-collinear points in it (plane ABC if A, B, C are not all on one line).

Visual Definitions with Simple Diagrams

Diagrams are models. They help you reason, but they are not the objects themselves. Use these visual cues to interpret what the diagram is communicating.

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What Each Term Means (and Does Not Mean)

Point: location without size

Even if a dot in a picture looks large, the geometric point has no size. The dot is just a symbol so your eyes can find the location.

Line: no thickness, no endpoints

A drawn line on paper has thickness and ends at the edge of the page. A geometric line has neither. The arrowheads are the diagram’s way of saying, “keep going.”

Plane: flat, infinite, and edge-free

The parallelogram shape is not the plane’s boundary. It is a window into the plane. Points may be shown on it, and lines may lie in it, but the plane continues beyond the drawn edges.

Vocabulary Box

Diagram-Reading Skills

1) Reading arrowheads

Arrowheads indicate that a line continues beyond what is drawn.

• Two arrowheads: a line (infinite in both directions).
• One arrowhead: often used for a ray (not the focus here, but you may see it).
• No arrowheads: could represent a segment or just a portion of a line; check labels and context.

2) Reading labels

Labels tell you what the diagram intends.

• Point labels: capital letters placed next to dots (e.g., A, B).
• Line labels: either two point names (line AB) or a lowercase letter near the line (line l).
• Plane labels: a capital letter written on the plane (plane P) or three non-collinear points (plane ABC).

3) Coplanar vs non-coplanar points

To decide whether points are coplanar, look for whether they can all be placed on the same drawn plane (the same parallelogram surface) or described as lying in the same plane label.

• Coplanar example: If points A, B, and C are drawn on plane P, then they are coplanar in plane P.
• Non-coplanar example: If A and B are on plane P, but D is drawn off that plane (floating above/below), then A, B, D are non-coplanar (they do not all lie in plane P).

4) Stating relationships clearly

Geometry writing is precise. Use phrases like these:

• Point A lies on line l.
• Points A and B determine line AB.
• Line AB lies in plane P.
• Points A, B, and C lie in plane P.

Step-by-Step: How to Identify Points, Lines, and Planes in a Diagram

Step 1: Find the plane (if shown)

Look for a slanted parallelogram and a label such as P. Everything drawn on that surface is intended to be in that plane unless the diagram indicates otherwise.

Step 2: Identify points

Locate dots and read their capital-letter labels. Each labeled dot is a point.

Step 3: Identify lines

Look for straight drawings with arrowheads. If a line passes through two labeled points, you can name it using those points (line AB).

Step 4: Decide “on,” “in,” or “through” relationships

• If a point is drawn directly on a line, say the point lies on the line.
• If a line is drawn entirely on the plane surface, say the line lies in the plane.
• If a point is drawn on the plane surface, say the point lies in the plane.

Practice Prompts (Diagram-Based)

Use the diagram descriptions below to answer. Write your answers in complete sentences using the vocabulary from this chapter.

Prompt 1

A slanted parallelogram represents plane P. Points A, B, and C are drawn on the plane. A straight line with arrowheads passes through A and B and is drawn entirely on the plane.

• List the points shown.
• Name the line shown using two points.
• State two true relationships (example format: Points A, B, and C lie in plane P.).

Prompt 2

Plane P is shown. Points A and B are on plane P. Point D is drawn above the plane (not on the parallelogram). A line with arrowheads passes through A and D.

• Are points A, B, and D coplanar? Explain using the diagram information.
• State whether D lies in plane P.
• Write a sentence naming the line through A and D.

Prompt 3

Two lines with arrowheads cross at point E. One line contains points E and F. The other contains points E and G. No plane is labeled.

• Name both lines using point labels.
• What is the intersection of the two lines?
• Write one sentence that uses the phrase lies on.