### Geometric Definitions

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.

When labeling rays, always write the endpoint under the side WITHOUT the arrow, as in \begin{align*}\overrightarrow{C D}\end{align*} or \begin{align*}\overleftarrow{D C},\end{align*}since that letter represents the end of the ray and the arrow indicates the direction that the ray continues.

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, whereas a theorem must be *proven* true. Proving theorems is the topic of another Concept.

**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.

#### Applying Definitions

What best describes San Diego, California on a globe? A point, a line, or a plane?

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

#### Interpreting Pictures

1. Use the picture below to answer these questions.

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

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

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

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

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

Yes

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

Yes

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

\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.

### Examples

#### Example 1

What best describes the surface of a movie screen? A point, a line, or a plane?

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

#### Example 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?

Neither

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

Yes

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

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

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

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

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*}

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. How do the figures below intersect?

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 along line \begin{align*}l.\end{align*} The last figure, three planes, intersect at point \begin{align*}S.\end{align*}

### Review

- 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 plot points on the coordinate plane and graph 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*}.

### Review (Answers)

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