Geometry Fundamentals: Undefined and Defined Terms

Conceptual Overview: Geometry is constructed upon a framework of three primary "undefined terms." These terms are accepted without formal definitions and serve as the essential foundation for all subsequent geometric concepts and constructions.
The Three Undefined Terms:
- Point
- Line
- Plane
Defined Terms: From these three undefined terms, mathematicians define other geometric figures, including line segments, rays, angles, collinear points, and coplanar points.
Learning Competency: The goal is to illustrate and describe concepts such as point, line, ray, line segment, angle, and plane using physical models and specific geometric notations.
Learning Objectives:
1. Describe the specific characteristics of the three undefined terms in geometry.
2. Define the various subsets of a line and the components of an angle.
3. Illustrate point, line, plane, ray, line segment, and angle using descriptive models.
4. Sketch and label point, line, plane, ray, line segment, and angle using standard geometric notation.
5. Identify and accurately name geometric figures based on provided diagrams.

Verbatim Definition: A point represents an exact location.
Dimension: Has no size, no dimension, and specifically has no length, width, or height.
Notation and Naming: Points are named using a single capital letter (e.g., Point $A$ , Point $M$ ).

Verbatim Definition: A line is made up of infinitely many points and extends endlessly in opposite directions.
Characteristics:
- Represents a straight path.
- Possesses no thickness or width.
- Extends infinitely in both ( $2$ ) directions.
- Visual representation includes arrowheads on both ends to denote infinite extension.
Notation and Naming: A line can be named by a single lowercase letter or by any two points residing on that line.

Verbatim Definition: A plane is a flat surface that extends infinitely in all directions.
Characteristics:
- It is a flat surface.
- It contains points and lines within its boundaries.
- It extends endlessly in all directions.
Notation and Naming: A plane is named by a single capital letter or by naming three non-collinear points that lie on the plane.

Ray:
- Definition: A part of a line that starts at one endpoint and extends infinitely in one direction.
- Characteristics: Features one endpoint and one arrowhead. It extends endlessly in only one direction.
- Naming Rule: The endpoint must be written first when using geometric notation to name a ray.
Line Segment:
- Definition: A part of a line bounded by two distinct endpoints.
- Characteristics: Features two clear endpoints and has a fixed length. Unlike a line or a ray, it cannot extend further than its endpoints.

Angle: A figure formed by two rays that share a common endpoint.
Vertex: The common endpoint where the two rays of an angle meet.
Sides of an Angle: The two rays that originate from the vertex to form the angle.
Naming Convention: When naming an angle using three letters, the vertex must always be the middle letter.

Foundational Logic: Undefined terms (point, line, plane) are used to describe all "defined terms."
Dimensionality:
- A point has zero ( $0$ ) dimensions.
- A line extends infinitely in two ( $2$ ) directions.
- A plane extends infinitely in all directions.
- A ray extends infinitely in only one ( $1$ ) direction.
Length and Constraint:
- A line segment has a fixed, measurable length.
- Lines and rays cannot be measured for total length because they extend infinitely.
Collinearity and Coplanarity:
- Points on the same line are collinear.
- Points on the same plane are coplanar.
Naming Requirements:
- Three non-collinear points are required to uniquely name a plane.
- The endpoint of a ray is always the first letter in its name.
- The vertex of an angle must be the central letter in its three-letter designation.
- The sides of an angle are classified specifically as rays.
Synthesis: Geometry starts with the three undefined terms (point, line, and plane); mastery of their characteristics, notations, and real-life models is essential for the further study of geometric figures like vertices, segments, and angles.