Geometry

Basics of Geometry

Big Picture

Geometry is founded upon some very important basic concepts. These include points, angles, lines, and line segments. There are many postulates that govern the way that we can use these basic concepts. After the basic foundation is created, we can progress to harder, more advanced geometry.

Key Terms

Undefined Term: term that cannot be mathematically defined using other known words. The undefined terms point, line, and plane are the building blocks of geometry.

Point: Location that has no size (no dimension).

Line: Infinite series of points in a row (1-dimensional). It has direction and location and is always straight.

Plane: Any flat, 2-dimensional surface.

Undefined terms can be used to define other geometric terms:

Segment: Portion of a line that is ended by two points.

Ray: Portion of line that has only one endpoint and extends infinitely in the other direction.

Endpoint: Point at the end of a segment or at the start of a ray.

Space: Set of all points expanding in three dimensions. It has no shape and no limits.

Collinear: Points that lie on the same line.

Coplanar: Points or lines that lie within the same plane.

Intersection: Point or set of points where lines, planes, segments, or rays cross.

Postulate: Basic rule that is assumed to be true. Also known as an axiom.

Drawing and Labeling

Term

Notation

Diagram

Point

A capital label (A, L, F)

Point

Line

A lowercase letter (line g) or two points on the line \(\overleftrightarrow{PQ} \text{or} \overleftrightarrow{QP}\)

Line

Plane

A script capital letter (plane M) or three points not on a line (plane ABC)

Think of a plane as a huge sheet of paper!

Plane

Segment

The two endpoints (\(\overline{AB}\) or \(\overline{BA}\))

Segment

Ray

Endpoint and any other point on the ray (\(\overrightarrow{CD}\))

Ray

Geometry

Basics of Geometry cont.

Geometric Figures

Everything in the figure below occupies 3-dimensional space.

Geometric Figures
  • The points A, B, and C are collinear.
  • Point  C lies on line h between A and B. A point is between two other points when they are in a straight line; G is not between A and B.
  • \(\overrightarrow{CA}\) and \(\overrightarrow{CB}\) are opposite rays that have a common endpoint and form a line.
  • Point G is not on line h.
  • Points A, G, and K and lines h and i are coplanar.
  • Point  E is not in plane j, but points E, A, and B are coplanar even though the plane is not drawn in.

When  drawing  or  labeling  geometric  figures,  be  very specific!

  • Include arrows to show that the line extends to infinity.
  • Label points.
  • When labeling rays, make sure that the end point is under the side without an arrow. \(\overrightarrow{KF}\) is NOT the same as \(\overrightarrow{FK}\)

Remember that arrows extend forever. In the figure below, even though the intersection is not shown, the two lines intersect at a point to the right.

coplanar

Basic Postulates

Postulates for Points, Lines, and Planes

  • There is exactly one line through any two points.

    Postulates for Points
  • There is exactly one plane that contains any three non-collinear points.

    Postulates for Planes
  • A line connecting points in a plane also lies within the plane.

    Postulates for Lines

Postulates for Intersection

  • The intersection of any two distinct lines will be a single point.

    Postulates for Intersection
  • The intersection of two distinct planes is a line

    intersection of two distinct planes