Precise definitions of geometric terms (point, line, plane, angle, circle)
Precise Geometric Definitions
Precise Geometric Definitions is the opening topic of Transformations, Congruence & Proof in the California Common Core State Standards. It is aligned to Standard G-CO.1, which requires students to know precise definitions of angle, circle, perpendicular line, parallel line, and line segment based on the undefined notions of point, line, and distance.
The precise definitions of point, line, plane, angle, and circle are built from the undefined notions of point, line, and distance along a line.
Every question with a fully worked solution.
- Geometric Terms Watch
Theory
Geometry begins with three undefined terms — point, line, and plane — accepted without definition and used to define everything else.
From them come precise definitions:
- Segment \(\overline{AB}\): the part of a line between two endpoints.
- Ray \(\overrightarrow{AB}\): a part of a line from one endpoint through another point, extending forever.
- Angle: two rays with a common endpoint (the vertex).
- Circle: all points a fixed distance (radius) from a center point.
Standard notation:
How to write a precise definition
- Start from the undefined terms or already-defined terms.
- State the category (a segment is part of a line).
- Add the distinguishing condition (between two endpoints).
- Check it is unambiguous — it should include exactly the intended figures.
Three terms are accepted without definition and used to define all others:
- point — a location with no size
- line — a straight, endless path of points
- plane — a flat surface extending forever
A segment \(\overline{AB}\) is the part of a line between two points \(A\) and \(B\), including both endpoints.
An angle is formed by two rays (the sides) sharing a common endpoint (the vertex). \(\angle ABC\) has vertex \(B\).
A circle is the set of all points in a plane a fixed distance (the radius) from a fixed point (the center).
Common pitfalls
Frequently asked questions
What are the undefined terms in geometry?
Point, line, and plane. They are accepted without definition and used to define all other geometric terms.
What is the definition of an angle?
Two rays that share a common endpoint, called the vertex. \(\angle ABC\) has its vertex at \(B\).
How is a circle defined?
The set of all points in a plane at a fixed distance, the radius, from a fixed center point.
What is the difference between a segment, a ray, and a line?
A segment has two endpoints, a ray has one endpoint and goes on forever, and a line extends forever in both directions.