What if you were given a picture of a figure or an object, like a map with cities and roads marked on it? How could you explain that picture geometrically? After completing this Concept, you'll be able to describe such a map using geometric terms.

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CK-12 Foundation: Chapter1BasicGeometricDefinitionsA

James Sousa: Definitions of and Postulates Involving Points, Lines, and Planes

### Guidance

A **point** is an exact location in space. A point describes a location, but has no size. Dots are used to represent points in pictures and diagrams. These points are said “Point \begin{align*}A\end{align*},” “Point \begin{align*}L\end{align*}”, and “Point \begin{align*}F\end{align*}.” Points are labeled with a CAPITAL letter.

A **line** is a set of infinitely many points that extend forever in both directions. A line, like a point, does not take up space. It has direction, location and is always straight. Lines are one-dimensional because they only have length (no width). A line can by named or identified using any two points on that line or with a lower-case, italicized letter.

This line can be labeled \begin{align*}\overleftrightarrow{P Q}, \ \overleftrightarrow{Q P}\end{align*} or just \begin{align*}g\end{align*}. You would say “line \begin{align*}PQ\end{align*},” “line \begin{align*}QP\end{align*},” or “line \begin{align*}g\end{align*},” respectively. Notice that the line over the \begin{align*}\overleftrightarrow{P Q}\end{align*} and \begin{align*}\overleftrightarrow{Q P}\end{align*} has arrows over both the \begin{align*}P\end{align*} and \begin{align*}Q\end{align*}. The order of \begin{align*}P\end{align*} and \begin{align*}Q\end{align*} does not matter.

A **plane** is infinitely many intersecting lines that extend forever in all directions. Think of a plane as a huge sheet of paper that goes on forever. Planes are considered to be two-dimensional because they have a length and a width. A plane can be classified by any three points in the plane.

This plane would be labeled Plane \begin{align*}ABC\end{align*} or Plane \begin{align*}\mathcal{M}\end{align*}. Again, the order of the letters does not matter.

We can use **point**, **line**, and **plane** to define new terms. **Space** is the set of all points extending in ** three** dimensions. Think back to the plane. It extended along two different lines: up and down, and side to side. If we add a third direction, we have something that looks like three-dimensional space, or the real-world.

Points that lie on the same line are **collinear**. \begin{align*}P, Q, R, S\end{align*}, and \begin{align*}T\end{align*} are collinear because they are all on line \begin{align*}w\end{align*}. If a point \begin{align*}U\end{align*} were located above or below line \begin{align*}w\end{align*}, it would be **non-collinear**.

Points and/or lines within the same plane are **coplanar**. Lines \begin{align*}h\end{align*} and \begin{align*}i\end{align*} and points \begin{align*}A, B, C, D, G\end{align*}, and \begin{align*}K\end{align*} are **coplanar** in Plane \begin{align*}\mathcal{J}\end{align*}. Line \begin{align*}\overleftrightarrow{KF}\end{align*} and point \begin{align*}E\end{align*} are **non-coplanar** with Plane \begin{align*}\mathcal{J}\end{align*}.

An **endpoint** is a point at the end of a line segment. Line segments are labeled by their endpoints, \begin{align*}\overline{AB}\end{align*} or \begin{align*}\overline{BA}\end{align*}. Notice that the bar over the endpoints has NO arrows. Order does not matter.

A **ray** is a part of a line with one endpoint that extends forever in the direction opposite that endpoint. A ray is labeled by its endpoint and one other point on the line.

Of lines, line segments and rays, rays are the only one where order matters. When labeling, always write the endpoint under the side WITHOUT the arrow, \begin{align*}\overrightarrow{C D}\end{align*} or \begin{align*}\overleftarrow{D C}\end{align*}.

An **intersection** is a point or set of points where lines, planes, segments, or rays cross each other.

##### Postulates

With these new definitions, we can make statements and generalizations about these geometric figures. This section introduces a few basic postulates. Throughout this course we will be introducing Postulates and Theorems so it is important that you understand what they are and how they differ.

**Postulates** are basic rules of geometry. We can assume that all postulates are true, much like a definition. **Theorems** are statements that can be proven true using postulates, definitions, and other theorems that have already been proven.

The only difference between a theorem and postulate is that a postulate is *assumed* true because it cannot be shown to be false, a theorem must be *proven* true. We will prove theorems later in this course.

**Postulate #1:** Given any two distinct points, there is exactly one (straight) line containing those two points.

**Postulate #2:** Given any three non-collinear points, there is exactly one plane containing those three points.

**Postulate #3:** If a line and a plane share two points, then the entire line lies within the plane.

**Postulate #4:** If two distinct lines intersect, the intersection will be one point.

**Postulate #5:** If two distinct planes intersect, the intersection will be a line.

When making geometric drawings, be sure to be clear and label all points and lines.

#### Example A

What best describes San Diego, California on a globe?

A. point

B. line

C. plane

Answer: A city is usually labeled with a dot, or point, on a globe.

#### Example B

Use the picture below to answer these questions.

a) List another way to label Plane \begin{align*}\mathcal{J}\end{align*}.

b) List another way to label line \begin{align*}h\end{align*}.

c) Are \begin{align*}K\end{align*} and \begin{align*}F\end{align*} collinear?

d) Are \begin{align*}E, B\end{align*} and \begin{align*}F\end{align*} coplanar?

Answer:

a) Plane \begin{align*}BDG\end{align*}. Any combination of three coplanar points that are not collinear would be correct.

b) \begin{align*}\overleftrightarrow{AB}\end{align*}. Any combination of two of the letters \begin{align*}A, B\end{align*}, or \begin{align*}C\end{align*} would also work.

c) Yes

d) Yes

#### Example C

Describe the picture below using all the geometric terms you have learned.

Answer:

\begin{align*}\overleftrightarrow{A B}\end{align*} and \begin{align*}D\end{align*} are coplanar in Plane \begin{align*}\mathcal{P}\end{align*}, while \begin{align*}\overleftrightarrow{B C}\end{align*} and \begin{align*}\overleftrightarrow{A C}\end{align*} intersect at point \begin{align*}C\end{align*} which is non-coplanar.

Watch this video for help with the Examples above.

CK-12 Foundation: Chapter1BasicGeometricDefinitionsB

### Vocabulary

A ** point** is an exact location in space. A

**is infinitely many points that extend forever in both directions. A**

*line***is infinitely many intersecting lines that extend forever in all directions.**

*plane***is the set of all points extending in three dimensions. Points that lie on the same line are**

*Space***. Points and/or lines within the same plane are**

*collinear***. An**

*coplanar***is a point at the end of part of a line. A**

*endpoint***is a part of a line with two endpoints. A**

*line segment***is a part of a line with one endpoint that extends forever in the direction opposite that point. An**

*ray***is a point or set of points where lines, planes, segments, or rays cross. A**

*intersection***is a basic rule of geometry is assumed to be true. A**

*postulate***is a statement that can be proven true using postulates, definitions, and other theorems that have already been proven.**

*theorem*### Guided Practice

1. What best describes the surface of a movie screen?

A. point

B. line

C. plane

2. Answer the following questions about the picture.

a) Is line \begin{align*}l\end{align*} coplanar with Plane \begin{align*}\mathcal{V}\end{align*}, Plane \begin{align*}\mathcal{W}\end{align*}, both, or neither?

b) Are \begin{align*}R\end{align*} and \begin{align*}Q\end{align*} collinear?

c) What point belongs to neither Plane \begin{align*}\mathcal{V}\end{align*} nor Plane \begin{align*}\mathcal{W}\end{align*}?

d) List three points in Plane \begin{align*}\mathcal{W}\end{align*}.

3. Draw and label the intersection of line \begin{align*}\overleftrightarrow{A B}\end{align*} and ray \begin{align*}\overrightarrow{C D}\end{align*} at point \begin{align*}C\end{align*}.

4. How do the figures below intersect?

**Answers:**

1. The surface of a movie screen is most like a plane.

2. a) Neither

b) Yes

c) \begin{align*}S\end{align*}

d) Any combination of \begin{align*}P, O, T\end{align*}, and \begin{align*}Q\end{align*} would work.

3. It does not matter the placement of \begin{align*}A\end{align*} or \begin{align*}B\end{align*} along the line nor the direction that \begin{align*} \overrightarrow{C D}\end{align*} points.

4. The first three figures intersect at a point, \begin{align*}P, Q\end{align*} and \begin{align*}R\end{align*}, respectively. The fourth figure, two planes, intersect in a line, \begin{align*}l\end{align*}. And the last figure, three planes, intersect at one point, \begin{align*}S\end{align*}.

### Interactive Practice

### Practice

- Name this line in two ways.

- Name the geometric figure below in two different ways.

- Draw three ways three different planes can (or cannot) intersect.
- What type of geometric object is made by the intersection of a sphere (a ball) and a plane? Draw your answer.

Use geometric notation to explain each picture in as much detail as possible.

For 6-15, determine if the following statements are ALWAYS true, SOMETIMES true, or NEVER true.

- Any two distinct points are collinear.
- Any three points determine a plane.
- A line is composed of two rays with a common endpoint.
- A line segment has infinitely many points between two endpoints.
- A point takes up space.
- A line is one-dimensional.
- Any four distinct points are coplanar.
- \begin{align*}\overrightarrow{A B}\end{align*} could be read “ray \begin{align*}AB\end{align*}” or “ray \begin{align*}BA\end{align*}.”
- \begin{align*}\overleftrightarrow{A B}\end{align*} could be read “line \begin{align*}AB\end{align*}” or “line \begin{align*}BA\end{align*}.”
- Theorems are proven true with postulates.

In Algebra you plotted points on the coordinate plane and graphed lines. For 16-20, use graph paper and follow the steps to make the diagram on the same graph.

- Plot the point (2, -3) and label it \begin{align*}A\end{align*}.
- Plot the point (-4, 3) and label it \begin{align*}B\end{align*}.
- Draw the segment \begin{align*}\overline{AB}\end{align*}.
- Locate point \begin{align*}C\end{align*}, the intersection of this line with the \begin{align*}x-\end{align*}axis.
- Draw the ray \begin{align*}\overrightarrow{CD}\end{align*} with point \begin{align*}D (1, 4)\end{align*}.