Lines and Planes

Hi, and welcome to this video on Lines and Planes! The study of geometry is very much language-based, meaning that there are countless terms, relationships, and figures with meanings that are dependent on an understanding of other concepts. It can get pretty confusing if the foundational terms are not understood. In this video, we’re going to start with the most basic figures: a point, a line, and a plane. These “undefined” terms are described, rather than being defined, and they support the definitions of all other geometric terms.

To start off, what is a point?

A point is described as a very specific location, or position, in a plane. The notation for a point is a dot, but that dot does not have any dimension (length, width, circumference). A point is named with a capital letter, as in “point A”

A line is described as a “path,” as if a point was dragged or is moving. A straight line extends infinitely in opposite directions. A line is typically named with a lowercase letter, or by referencing two points on the line, with a line symbol above. The line notation has arrows on either end to indicate that they extend forever. Points that lie on a line are referred to as collinear.

A plane surface, has length and width, and extends infinitely in all directions. A flat surface, like a wall, floor, or ceiling, can be imagined as finite planes where geometric figures, like points and lines, can be drawn. A plane is typically named with a letter in script or italics (plane m) or by naming three points that lie on the plane, (plane ABC). Using three points in the naming of a plane lends to the perception of a two-dimensional surface. Points that lie in the same plane are said to be coplanar.

Planes that intersect do so at a line, and it is possible for three planes to intersect at exactly one point.

Now that we know these basic components, we can build our knowledge with terms that incorporate them in their definitions. For example:

A line segment is the portion of a line that lies between two points on the line. The two points are called endpoints, and are included in the line segment, as are all the points that are between them. A line segment with endpoints A and B would be referenced as \(\overline{AB}\).

A ray starts at one point and extends infinitely in one direction on a plane. The ray symbol has one arrow indicating the starting point and the direction of the ray.

When two lines on a plane cross each other, they are referred to as intersecting lines. Intersecting lines on a plane cross at exactly one point.

Because a line segment has length that can be measured between the endpoints, the exact midpoint of the segment can be determined. A point, line, or ray, or plane that crosses a line segment at the midpoint is called a bisector.

Intersecting lines on a plane that cross at 90° angles, or “right angles,” are perpendicular to each other. Examples of perpendicular lines can be found on window panes, or on door frames.

Lines on a plane that never cross are called parallel. These lines are exactly the same distance apart at all points, like the double yellow lines on a road, or tire tracks of a car.

A line that crosses two lines in a plane at two distinct points is called a transversal line. Transversal lines in combination with special angle relationships are used to determine whether lines in a plane are parallel.

As you can see, it is essential to understand the relationships between the “undefined” terms of a point, a line and a plane in order to strengthen and expand your understanding of other geometry concepts. It’s important to review these frequently from the ground up to keep pace and to retain your knowledge.

Thanks for watching, and happy studying!

Practice Questions

Question #1:

 
A ________ is a part of a line that has one fixed starting point, and extends infinitely in one direction.

Midpoint

Ray

Line Segment

Transversal

Answer:

A ray is a part of a line that has one fixed starting point, and extends infinitely in one direction.
Example of a Ray

Question #2:

 
Name the plane in the image below.
Plane with points EFG and a T

Plane EF

Shape T

Plane G

Plane EFG or Plane T

Answer:

A plane can be named by an italicized letter such as T, or by three non-collinear points that lie on the plane, such as EFG.

Question #3:

 
Which capital letters in the alphabet have parallel lines.
picture showing the alphabet

E, F, H, M, N, X, Y, and Z

E, F, H and M

E, H, M, N, and Z

E, F, H, M, N, W, and Z

Answer:

Parallel lines will never cross each other even when extended infinitely. The letters E, F, H, M, N, W, and Z consist of parallel lines.

Question #4:

 
Which geometric term best describes how the city of Austin, Texas would be represented on a globe?

Line

Plane

Point

Ray

Answer:

A point describes a location, such as Austin, Texas on a globe. In geometry, a point does not take up space, but in pictures or diagrams they are drawn as dots.

Question #5:

 
Andrew wants to build a model of a skyscraper using paper. He decides to design the building as a triangular prism. How many planes will be used to create this model?
picture of a skyscraper
paper model of a triangular pyramid skyscraper

Three planes

Four planes

Five planes

Six planes

Answer:

The model will be built from five planes: top, bottom, and three sides.

 

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by Mometrix Test Preparation | Last Updated: November 17, 2021