What if you were given a picture of a figure or 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.

### Watch This

CK-12 Basic Geometric Definitions - Guidance

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

### Guidance

A **point** is an exact location in space. It describes a location, but has no size. Examples are shown below:

Label It |
Say It |
---|---|

\begin{align*}A\end{align*} |
point \begin{align*}A\end{align*} |

A **line** is infinitely many points that extend forever in both directions. Lines have direction and location and are *always straight*.

Label It |
Say It |
---|---|

line \begin{align*}g\end{align*} |
line \begin{align*}g\end{align*} |

\begin{align*}\overleftrightarrow{\text{PQ}}\end{align*} |
line \begin{align*}PQ\end{align*} |

A **plane** is a flat surface that contains infinitely many intersecting lines that extend forever in all directions. Think of a plane as a huge sheet of paper with no thickness that goes on forever.

Label It |
Say It |
---|---|

Plane \begin{align*}\mathcal{M}\end{align*} |
Plane \begin{align*}M\end{align*} |

Plane \begin{align*}ABC\end{align*} |
Plane \begin{align*}ABC\end{align*} |

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 in two dimensions, what we think of as up/down and left/right. If we add a third dimension, one that is perpendicular to the other two, we arrive at three-dimensional space.

Points that lie on the same line are **collinear**. \begin{align*}P, Q, R, S\end{align*}**non-collinear**.

Points and/or lines within the same plane are **coplanar**. Lines \begin{align*}h\end{align*}**coplanar** in Plane \begin{align*}\mathcal{J}\end{align*}**non-coplanar** with Plane \begin{align*}\mathcal{J}\end{align*}

An **endpoint** is a point at the end of a line segment. A **line segment** is a portion of a line with two endpoints. Or, it is a finite part of a line that stops at both ends. Line segments are labeled by their endpoints. Order does not matter.

Label It |
Say It |
---|---|

\begin{align*}\overline{AB}\end{align*} |
Segment \begin{align*}AB\end{align*} |

\begin{align*}\overline{BA}\end{align*} |
Segment \begin{align*}BA\end{align*} |

A **ray** is a part of a line. It begins with an endpoint and extends forever away from the endpoint in one direction, perfectly straight. A ray is labeled by its endpoint and one other point on the line. A ray is labeled by its endpoint and one other point on the ray. For rays, order does matter. When labeling, put the endpoint under the side WITHOUT the arrow.

Label It |
Say It |
---|---|

\begin{align*}\overrightarrow{CD}\end{align*} |
Ray \begin{align*}CD\end{align*} |

\begin{align*}\overleftarrow{DC}\end{align*} |
Ray \begin{align*}CD\end{align*} |

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

**Postulates**

A **postulate** is a basic rule of geometry. Postulates are assumed to be true (rather than proven), much like definitions. The following is a list of some basic postulates.

**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.

Lines \begin{align*}l\end{align*}

**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*}

d) Are \begin{align*}E, B\end{align*}

Answer:

a) Plane \begin{align*}BDG\end{align*}

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 be correct.

c) Yes, they both lie on \begin{align*}\overleftrightarrow{KF}\end{align*}.

d) Yes, even though \begin{align*}E\end{align*} is not in Plane \begin{align*}\mathcal{J}\end{align*}, any three points make a distinct plane. Therefore, the three points create Plane \begin{align*}EBF\end{align*}.

#### Example C

What best describes a straight road that begins in one city and stops in a second city?

A. ray

B. line

C. segment

D. plane

Answer: The straight road connects two cities, which are like endpoints. The best term is segment, or \begin{align*}C\end{align*}.

CK-12 Basic Geometric Definitions E

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### 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 a figure matching the following description: Line \begin{align*}\overleftrightarrow{AB}\end{align*} and ray \begin{align*}\overrightarrow{CD}\end{align*} intersect at point \begin{align*}C\end{align*}. Then, redraw so that the figure looks different but is still true to the description.

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

**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. Neither the position of \begin{align*}A\end{align*} or \begin{align*}B\end{align*} on the line, nor the direction that \begin{align*}\overrightarrow{CD}\end{align*} points matter.

For the second part, this is one way to draw the diagram differently:

4. \begin{align*}\overleftrightarrow{AB}\end{align*} and \begin{align*}D\end{align*} are coplanar in Plane \begin{align*}\mathcal{P}\end{align*}, while \begin{align*}\overleftrightarrow{BC}\end{align*} and \begin{align*}\overleftrightarrow{AC}\end{align*} intersect at point \begin{align*}C\end{align*}.

### Explore More

For questions 1-5, draw and label a figure to fit the descriptions.

- \begin{align*}\overrightarrow{CD}\end{align*} intersecting \begin{align*}\overline{AB}\end{align*} and Plane \begin{align*}P\end{align*} containing \begin{align*}\overline{AB}\end{align*} but not \begin{align*}\overrightarrow{CD}\end{align*}.
- Three collinear points \begin{align*}A, B\end{align*}, and \begin{align*}C\end{align*} and \begin{align*}B\end{align*} is also collinear with points \begin{align*}D\end{align*} and \begin{align*}E\end{align*}.
- \begin{align*}\overrightarrow{XY}, \overrightarrow{XZ}\end{align*}, and \begin{align*}\overrightarrow{XW}\end{align*} such that \begin{align*}\overrightarrow{XY}\end{align*} and \begin{align*}\overrightarrow{XZ}\end{align*} are coplanar, but \begin{align*}\overrightarrow{XW}\end{align*} is non-coplanar with both of the other rays.
- Two intersecting planes, \begin{align*}\mathcal{P}\end{align*} and \begin{align*}\mathcal{Q}\end{align*}, with \begin{align*}\overline{GH}\end{align*} where \begin{align*}G\end{align*} is in plane \begin{align*}\mathcal{P}\end{align*} and \begin{align*}H\end{align*} is in plane \begin{align*}\mathcal{Q}\end{align*}.
- Four non-collinear points, \begin{align*}I, J, K\end{align*}, and \begin{align*}L\end{align*}, with line segments connecting all points to each other.
- Name this line in five ways.
- Name the geometric figure in three different ways.
- Name the geometric figure below in two different ways.
- What is the best possible geometric model for a soccer field? Explain your answer.
- List two examples of where you see rays in real life.
- What type of geometric object is the intersection of a line and a plane? Draw your answer.
- What is the difference between a postulate and a theorem?

For 13-16, use geometric notation to explain each picture in as much detail as possible.

For 17-25, determine if the following statements are true or false.

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

### Answers for Explore More Problems

To view the Explore More answers, open this PDF file and look for section 1.1.