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 |
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point |

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 | line |

line |

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.

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

Plane | Plane |

Plane | Plane |

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**. , and are collinear because they are all on line . If a point were located above or below line , it would be **non-collinear**.

Points and/or lines within the same plane are **coplanar**. Lines and and points , and are **coplanar** in Plane . Line and point are **non-coplanar** with Plane .

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 |
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Segment | |

Segment |

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 ray. For rays, order matters. When labeling, put the endpoint under the side WITHOUT the arrow.

Label It |
Say It |
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Ray | |

Ray |

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

##### 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 and intersect at 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 .

b) List another way to label line .

c) Are and collinear?

d) Are and coplanar?

Answer:

a) Plane . Any combination of three coplanar points that are not collinear would be correct.

b) . Any combination of two of the letters , or would also work.

c) Yes

d) Yes

#### Example C

What best describes a straight road connecting two cities?

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 .

CK-12 Basic Geometric Definitions E

### 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 coplanar with Plane , Plane , both, or neither?

b) Are and collinear?

c) What point belongs to neither Plane nor Plane ?

d) List three points in Plane .

3. Draw and label a figure matching the following description: Line and ray intersect at point . 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)

d) Any combination of , and would work.

3. Neither the position of or on the line, nor the direction that points matter.

For the second part:

4. and are coplanar in Plane , while and intersect at point .

### Practice

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

- intersecting and Plane containing but not .
- Three collinear points , and and is also collinear with points and .
- , and such that and are coplanar, but is not.
- Two intersecting planes, and , with where is in plane and is in plane .
- Four non-collinear points, , and , 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.
- could be read “ray ” or “ray “.”
- could be read “line ” or “line .”