Undefined Terms of Geometry

Point, Line, Plane

The formal study of geometryrequires simple ideas and

statements that can be acceptedas true without proof. The

undefined terms point, line, andplane provide the simple ideas.Basic postulates about points,

lines and planes can be acceptedwithout proof. These form the

building blocks for the firsttheorems that you can prove.

Two fundamental objects in geometry are points and lines. Even though we are familiar with these objects, we find it difficult to describe them due to their undefined nature.

"It might be important to expand on what is meant by undefined. We do not mean that any word that is undefined is understood by everyone, or that everyone will attach the same meaning to the word.

• Essentially, we mean that such a word is uninterpreted. In various situations different interpretations can be given to the undefined word. For example, 'line' may be interpreted – as a pencil mark drawn along the edge of a ruler– as a line of sight for a surveyor– as a string held tight between two people.

• What we are trying to do in these examples is to point out that 'line' in a given context has a meaning, or interpretation, associated with it. It's a bit like trying to describe the taste of salt; we aren't quite sure how to describe it and to each person it might taste a little different.

•

Points are labeled with capital letters, such as P.

• Point:  One of the basic undefined terms of geometry. A point has no length, width, or thickness, and we often use a dot to represent it. So in essence, can you see it? How do you know it is there?

• Can you see the wind?

• P

What are some real life examples that model a point?

Line:  One of the basic undefined terms of geometry. A line is made of infinite points. A line has no thickness but its length goes on forever in

two directions.

What are some real life examples that model a line?

Using Undefined terms and definition• A line extends in one

dimension. It is usually represented by a straight line with two arrowheads to indicate that the line extends without end in two directions. In this book, lines are always straight lines.

A

B

l

Line l or AB

Notice how Line AB is labeled.

Using Undefined terms and definition A plane extends in two

dimensions. It is usually represented by a shape that looks like a tabletop or wall. You must imagine that the plane extends without end even though the drawing of a plane appears to have edges.

A

BC

M

Plane M or plane ABC

A plane is often represented by a four-sided figure and can be named by a capital script letter or by three noncollinear

points (points that do not lie on the same line) on the plane.

What are some real life examples that model a plane?

The figure below shows two planes, A and B, that intersect. The points they have in common form a

line (line KL). Try to draw this on your paper.

Geogebra Webpage

Coplanar:  Set of points, lines, rays, line

segments, etc., that lie in the same plane.

• In plane R, points P, Q, and S are coplanar

l

Line l is not coplanar to points P,Q,S. Only one point on line l is in plane R.

Sketching intersections

• Sketch the figure described.

a. A line that intersects a plane in one point

Draw a plane and a line.

Emphasize the point where they meet.

Dashes indicate where the line is hidden by the plane

Sketching intersections

• Sketch the figure described.

b. Two planes that intersect in a line

Draw two planes. Emphasize the line

where they meet. Dashes indicate where

one plane is hidden by the other plane.