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!

Frequently Asked Questions

Q

What are lines?

A

In the context of mathematics, a line is an infinitely long collection of points. A line has no width or depth*, and it will continue to run in opposite directions forever. We designate that something is a line by marking arrows on both (visible) ends of a line segment.
*Because a line only has length as a dimension, it is a 1-dimensional object.

This:
short line segment
would not necessarily be considered a line in a math course because we don’t know if this object has specific endpoints or if it runs on forever.

However, we can turn this into a line by strategically placing 2 arrows:
line with arrows on either end
As you might guess, a line never has a visible ‘ending.’ In fact, a line doesn’t have any ending because it is infinitely long!

Q

How do you define a point?

A

A point is a set position (or “coordinate”) within a space. Traditionally, a point is designated by a simple ‘dot’ on some surface; and it’s either named by a single letter and/or described in \((x,y)\)– or \((x,y,z)\)-coordinate form.

Remember that a point is a dimensionless object because it doesn’t have any width, length, or depth.
on the left is point G, on the right is a coordinate graph with the point (2, 3) plotted and labeled

Q

What is a collinear point?

A

Collinear points are points that sit on the same line, line segment, or ray.
The points A, B, and C are collinear.
line with points A, B, and C
The points X and Y are not collinear because Y isn’t on the same line as X.line with point X in the middle and point Y to the right, off of the line

Q

What is a plane in math?

A

Think of a plane as the surface of an ever-lasting piece of paper: a flat surface that you can only move up and down or right and left on. You couldn’t move “in” or “out.”
parallelogram, dashed line going through the middle in both directions, vertical and horizontal
A plane is the collection of an infinite amount of points and lines, and it has both length and width (but no depth). Hence, a plane has only 2 dimensions.

We usually call it ‘the \(xy\)-plane,’ because we describe points and lines in relation to the horizontal axis, \(x\), and the vertical axis, \(y\).
coordinate graph with x-axis and y-axis labeled

Q

How do you identify a plane?

A

There are a few different ways to identify/construct a plane:

  • Using 3 non-collinear points: When looking at a collection of 3 points that don’t sit on a line together, one might notice that the only way to “connect” the points to one another is to have them sitting on the same plane together.
  • points on left I, II, and III, arrow pointing to the right, on the right is a parallelogram with points I, II and III in it

  • Using a line and a separate point: This method is very similar to the one outlined above. Imagine that we drew a line connecting 2 out of the 3 points from above; the only way to connect the two new objects would, again, be to draw a plane.
  • Line D and point Q with a parallelogram around them

  • Using a pair of intersecting lines: Imagine that you have two lines that cross over one another at some shared point. There is only one way to sit these lines on the same flat surface, and so we construct a plane that gives the two lines ‘common ground.’Line N and M intersecting and are surrounded by a parallelogram
  • Using a pair of parallel lines: Once again, this is similar to the intersecting lines method we just discussed. There is only one way to set up a plane for these parallel lines to sit on together.
  • two parallel horizontal lines labeled "Line O" and "line P" with a parallelogram around them

Q

Does a plane always have 3 points?

A

Technically, yes- a plane always has at least 3 points; because a plane is a collection of infinitely-many points. However, we can’t identify or construct a plane given less than 3 points.

Let’s say that we’ve been given the point A,
point labeled "A", and are told to, “find the unique plane that this point sits on.” Unfortunately, this is an impossible task! The amount of possible planes that we could come up with is endless.
rectangles in 3 different directions with point A in the center

Now let’s say that we’ve been given the points E and F
point E on the left and point F on the right
and told to find the plane (like above). While there’s only 1 unique line that connects this pair, again: we run into the problem that there are infinitely-many possible planes that the two points could be sitting on together.
three rectangles slightly rotated for each one, two points in the middle labeled "E" and "F"
When you are given 3 points, you have made certain that the space you are looking at is a plane because there is only one unique plane that all 3 points can lie on

Q

Does one line define a plane?

A

No, a single line cannot be used to define a unique plane. As mentioned above, 1 line can sit on a countless amount of possible planes.
three rectangles in three different directions with a line going through all of them labeled "line J"

Q

What are line segments?

A

It might seem silly at first, but a line segment is actually quite different from a line in math. This distinction is important: while a line continues infinitely in both directions, a line segment has a finite length. More specifically, line segments run from one “endpoint” to another, and these endpoints are the points that sit on both ends of the line segment.
line segment with two points on the end

Q

What does a ray look like in math?

A

A ray is kind of like the combination of a line and a line segment; it has 1 endpoint, but on the opposite end it continues on forever.
ray with a point on the left and an arrowhead on the right

Q

What is the difference between intersecting lines and perpendicular lines?

A

While 2 lines are considered intersecting lines if they cross over one another at a particular point, they are only considered perpendicular to one another if all 4 angles formed at the intersection point are right angles (each measures 90°).

Keep in mind: all perpendicular lines are intersecting lines, but not all intersecting lines are perpendicular.
2 intersecting lines and 2 perpendicular lines

The lines A and B are simply intersecting. However, the lines C and D are perpendicular.

Q

What are parallel and intersecting lines?

A

There are only 2 possible relationships that a pair of lines can have between themselves: they are either parallel or they are intersecting (or will eventually intersect) one another.

If it’s impossible for 2 lines to ever cross over one another, they are considered parallel. If the lines cross over one another at some point (we call this point the “intersection point”), we call them “intersecting lines.”
group of parallel lines and a group that will eventually intersect

The lines K and L are parallel to one another; and while K’ and L’ are not yet intersecting, they will eventually meet at the intersection point to the right.

Q

How do you identify a transversal line?

A

A transversal line is one which intersects at least 2 other lines. In the following figures, the dashed line is the transversal:
different groups of lines with a dotted line that intersects them

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 | This Page Last Updated: June 28, 2022