# 1.1: Points, Lines, and Planes

**At Grade**Created by: CK-12

## Learning Objectives

- Understand the terms
*point*,*line*, and*plane*. - Draw and label terms in a diagram.

## Review Queue

- List and draw pictures of five geometric figures you are familiar with.
- What shape is a yield sign?
- Solve the algebraic equations.
- \begin{align*}4x-7=29\end{align*}
- \begin{align*}-3x+5=17\end{align*}

**Know What?** Geometry is everywhere. Remember these wooden blocks that you played with as a kid? If you played with these blocks, then you have been “studying” geometry since you were a child.

How many sides does the octagon have? What is something in-real life that is an octagon?

**Geometry:** The study of shapes and their spatial properties.

## Building Blocks

**Point:** An exact location in space.

A point describes a **location**, but has no size. Examples:

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

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

**Line:** Infinitely many points that extend forever in both directions.

A line has **direction** and **location** is 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*} |

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

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

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

**Example 1:** What best describes San Diego, California on a globe?

A. point

B. line

C. plane

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

**Example 2:** What best describes the surface of a movie screen?

A. point

B. line

C. plane

**Solution:** The surface of a movie screen is most like a plane.

**Beyond the Basics** Now we can use **point**, **line**, and **plane** to define new terms.

**Space:** The set of all points expanding in ** three** dimensions.

Think back to the plane. It goes up and down, and side to side. If we add a third direction, we have space, something three-dimensional.

**Collinear:** Points that lie on the same line.

\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*} was above or below line \begin{align*}w\end{align*}, it would be **non-collinear**.

**Coplanar:** Points and/or lines within the same plane.

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*}.

**Example 3:** Use the picture above 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?

**Solution:**

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, C\end{align*} or \begin{align*}B\end{align*} would also work.

c) Yes

d) Yes

**Endpoint:** A point at the end of a line.

**Line Segment:** A line with two endpoints. Or, 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*} |

**Ray:** A line with one endpoint and extends forever in the other direction.

A ray is labeled by its endpoint and one other point on the line. For rays, order matters. When labeling, put endpoint under the side WITHOUT an 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*} |

**Intersection:** A point or line where lines, planes, segments or rays cross.

**Example 4:** What best describes a straight road connecting two cities?

A. ray

B. line

C. segment

D. plane

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

**Example 5:** Answer the following questions about the picture to the right.

a) Is line \begin{align*}l\end{align*} coplanar with Plane \begin{align*}\mathcal{V}\end{align*} or \begin{align*}\mathcal{W}\end{align*}?

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

c) What point is non-coplanar with either plane?

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

**Solution:**

a) No.

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.

**Further Beyond** This section introduces a few basic postulates.

**Postulates:** Basic rules of geometry. ** We can assume that all postulates are true**.

**Theorem:** A statement that is ** proven true** using postulates, definitions, and previously proven theorems.

**Postulate 1-1:** There is exactly one (straight) line through any two points.

**Investigation 1-1: Line Investigation**

- Draw two points anywhere on a piece of paper.
- Use a ruler to connect these two points.
- How many lines can you draw to go through these two points?

**Postulate 1-2:** One plane contains any three non-collinear points.

**Postulate 1-3:** A line with points in a plane is also in that plane.

**Postulate 1-4:** The intersection of two lines will be one point.

Lines \begin{align*}l\end{align*} and \begin{align*}m\end{align*} intersect at point \begin{align*}A\end{align*}.

**Postulate 1-5:** The intersection of two planes is a line.

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

**Example 6a:** Draw and label the intersection of line \begin{align*}\overleftrightarrow{AB}\end{align*} and ray \begin{align*}\overrightarrow{CD}\end{align*} at point \begin{align*}C\end{align*}.

**Solution:** It does not matter where you put \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.

**Example 6b:** Redraw Example 6a, so that it looks different but is still true.

**Solution:**

**Example 7:** Describe the picture below using the geometric terms you have learned.

**Solution:** \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*} which is non-coplanar.

**Know What? Revisited** The octagon has 8 sides. In Latin, “octo” or “octa” means 8, so octagon, literally means “8-sided figure.” An octagon in real-life would be a stop sign.

## Review Questions

- Questions 1-5 are similar to Examples 6a and 6b.
- Questions 6-8 are similar to Examples 3 and 5.
- Questions 9-12 are similar to Examples 1, 2, and 4.
- Questions 13-16 are similar to Example 7.
- Questions 17-25 use the definitions and postulates learned in this lesson.

For questions 1-5, draw and label an image 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 not.
- 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*}.”

## Review Queue Answers

- Examples could be triangles, squares, rectangles, lines, circles, points, pentagons, stop signs (octagons), boxes (prisms), or dice (cubes).
- A yield sign is a triangle with equal sides.
- \begin{align*}4x-7 = 29\!\\ {\;}\quad \ 4x = 36\!\\ {\;}\quad \ \ x = 9\end{align*}
- \begin{align*}-3x+5 =17\!\\ {\;}\quad \ -3x = 12\!\\ {\;}\qquad \ \ x = -4\end{align*}