1.6: Defined Terms
Learning Objectives
- Identify points/lines that are collinear, coplanar, non-collinear, and non-coplanar.
- Define and name segments and rays
Defined Terms: Points, Lines, and Planes
Now we can use point, line, and plane to define new terms. One word that has already been used is space. Space is the set of all points expanding in three dimensions.
- Space is the set of all ___________________________ expanding infinitely in three dimensions.
The prefix “co-” means "with".
Some other words that use “co-” are: co-pilot (pilot something with someone else), cooperate (operate with someone else), and co-exist (exist with someone else).
Can you think of other words that use the prefix “co-”?
Collinear and Non-Collinear
Points are said to be collinear if they lie along the same line. The picture below shows points \begin{align*}F, G,\end{align*} and \begin{align*}H\end{align*} are collinear. Point \begin{align*}J\end{align*} is non-collinear with the other three since it does not lie in the same line:
- Collinear points lie on the _______________________ line.
- Non-collinear points do not lie on the same _______________________.
Reading Check:
1. Which points are collinear in the picture below?
\begin{align*}{\;}\end{align*}
\begin{align*}{\;}\end{align*}
2. Which points are non-collinear?
Coplanar and Non-Coplanar
Similarly, points and lines can be coplanar if they lie within the same plane. The diagram below shows two lines ( \begin{align*}\overleftrightarrow{RS}\end{align*} and \begin{align*}\overleftrightarrow{TV}\end{align*}) and one point \begin{align*}(Q)\end{align*} that are coplanar. It also shows line \begin{align*}\overleftrightarrow{WX}\end{align*} and point \begin{align*}Z\end{align*} that are non-coplanar with \begin{align*}\overleftrightarrow{RS}\end{align*} and \begin{align*}Q\end{align*}:
- Points and lines are coplanar when they lie in the same ____________________.
Reading Check:
1. Look around the space you are in right now.
What planes do you see? What points lie in those planes?
\begin{align*}{\;}\end{align*}
\begin{align*}{\;}\end{align*}
\begin{align*}{\;}\end{align*}
\begin{align*}{\;}\end{align*}
2. The ground or floor underneath you is a plane.
What is a point that is non-coplanar with the ground or the floor?
\begin{align*}{\;}\end{align*}
\begin{align*}{\;}\end{align*}
\begin{align*}{\;}\end{align*}
\begin{align*}{\;}\end{align*}
Segments, Rays, and Endpoints
A segment designates a portion of a line that has two endpoints. Segments are named by their endpoints. Segments can also be named in any order, so the segment below could be named \begin{align*}\overline{AB}\end{align*} or \begin{align*}\overline{BA}\end{align*}:
- A line segment has two ______________________________, which is how the segment is named.
A ray is a portion of a line that has only one endpoint and extends infinitely in the other direction. Rays are named by their endpoints and another point on the line. The endpoint always comes first in the name of a ray, so we would write \begin{align*}\overrightarrow{GH}\end{align*} for the figure below:
- A ray has one ______________________ and extends ________________________ in the other direction.
- A ray is named by two points on it, with the endpoint ________________________.
Reading Check
1. Name each of the following figures:
2. Which points are collinear with point \begin{align*}T\end{align*} in the figure below?
3. Which lines, segments, or rays are coplanar with line \begin{align*}\overleftrightarrow{AB}\end{align*} in the figure below?
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