Lesson 1: Points, Lines, and Planes

Points, Lines, and Planes

Euclidean Geometry and its most basic parts

Point

A point is denoted by a single dot. The dot represents a precise location or space on a plane. A point has no dimensions because it is a location, not a thing.

Line

A line is made up of infinite points extending in opposite directions. They are usually named using two points on the line (i.e. line AB) or given a letter name in lowercase italics (i.e. line a). A line has 1 dimension: length.

Plane

A plane is composed of 3 or more noncollinear points. This means that the points cannot all be on a single straight line. They are usually named by the points they contain (i.e. plane ABC) or by an uppercase or lowercase letter (i.e. plane m). A plane extends infinitely in all directions and has 2 dimensions: length and width.

Naming Lines, Segments, Rays, and Planes

There are many names for the line that contains the points A, B, and C. We can use AB, BA, BC, AC, and CB to name a few. We can also call the plane in the diagram plane J, plane BCD, plane ACD, or plane ADG among others. Notice that when naming the plane, did not use collinear points to name the plane. This helps avoid confusion when discussing lines and planes simultaneously.

Multiple Choice

Name the line which contains points D and C.

• m

• AC

• EC

• CB

Open Ended

Name the plane in the diagram.

Ray

Opposite Rays

NOTE

When reading a diagram, you cannot assume any information that is not explicitly drawn or written within the diagram itself.

Collinear and Coplanar Points and Lines

Collinear points are points which lie on the same line. If a given diagram does not include those points on a line, you cannot assume they are collinear even if you could draw a line through them.

Points H and E are collinear points and lie on the line HE. Points D, E, and F are collinear and lie on the line DF. Points G and E are not collinear since no line is drawn through these two points in the diagram.

Collinear and Coplanar Points and Lines

Coplanar points are points which lie on the same plane. If a given diagram does not include those points on a plane, you cannot assume they are coplanar even if you could draw a plane that contains all of them.

Points D, E, and G are all coplanar and lie on plane DGF. Points H and E are not coplanar since there is no plane in the diagram which contains both of those points.

Collinear and Coplanar Points and Lines

Coplanar lines are lines which lie on the same plane.

The diagram to the right does not have any coplanar lines since there are no two or more lines drawn on plane DGF.

Multiple Choice

True or False: $\overrightarrow{AC}$  and  $\overrightarrow{CB}$  are opposite rays.

• True

• False

Multiple Select

Which of the following points are collinear?

• F

• B

• D

• A

• G

Multiple Select

Which of the following lines are coplanar?

• line AC

• line CD

• line KF

Multiple Choice

True or False: point F lies on plane J.

• True

• False

Intersecting Lines

The intersection of one or more lines is a point.

Intersecting Planes

The intersection of one or more planes makes a line.

Intersection of a line and a plane

The intersection of a plane and a line not on that plane is a point.

Multiple Choice

Name the intersection of line HE and plane DGF.

• line EF

• E

• line HE

• line GE

Multiple Choice

Name the intersection of planes M and N.

• line GE

• C

• line BC

• M x N