# 1.7: How Lines and Planes Intersect

Difficulty Level: At Grade Created by: CK-12

## Learning Objectives

• Explain how lines, planes, and a line & a plane intersect.
• Explain the difference between postulates and theorems.
• Identify and apply basic postulates of points, lines, and planes.

## Intersection

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

In the image above, R\begin{align*}R\end{align*} is the point of intersection of QR\begin{align*}\overrightarrow{QR}\end{align*} and SR\begin{align*}\overrightarrow{SR}\end{align*}. T\begin{align*}T\end{align*} is the intersection of MN\begin{align*}\overleftrightarrow{MN}\end{align*} and PO\begin{align*}\overleftrightarrow{PO}\end{align*}.

• An intersection is a point where objects ___________________ each other.

## Basic Postulates and Theorems

Now that we have some basic vocabulary, we can talk about the rules of geometry. Logical systems like geometry start with basic rules, and we call these basic rules postulates. We assume that a postulate is true and by definition a postulate is a statement that does not have to be proven.

• A postulate is a statement that does not have to be _________________________.

A theorem is a statement that can be proven true using postulates, definitions, logic, and other theorems we have already proven.

• A theorem is a statement that must be ___________________________.

This section introduces a few basic postulates that you must understand as you move on to learn other theorems. Some postulates and theorems have names; others do not.

1. True or False: When a line and a plane intersect, they meet at a point.

Why did you choose True or False? Explain your reasoning:

\begin{align*}{\;}\end{align*}

\begin{align*}{\;}\end{align*}

\begin{align*}{\;}\end{align*}

\begin{align*}{\;}\end{align*}

2. True or False: It is possible to have a plane with just three collinear points.

Why did you choose True or False? Explain your reasoning:

\begin{align*}{\;}\end{align*}

\begin{align*}{\;}\end{align*}

\begin{align*}{\;}\end{align*}

\begin{align*}{\;}\end{align*}

3. Fill in the blanks:

A __________________________ is a statement that is accepted without proof.

## Graphic Organizers for Lesson 4

Postulate or Theorem
Postulate Theorem
Description
• Assumed to be true
• Doesn’t have to be proven
Description
• Can’t assume that it’s true
• Must be proven
Examples
• A person can only be in one place at one time.
• A fish cannot live without water.
• A line must contain at least two points.
Examples
• Lisa was at home at 8:00 last night.
• Henry drinks eight glasses of water each day.
• The angles in a triangle add up to 180\begin{align*}180^\circ\end{align*}.

Add your own examples here. What are some things that are so obvious that they don’t have to be proven?

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Add your own examples here. What are some things you know are true, but you’d have to prove them to someone else?

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Line Postulate and Plane Postulates Try to disprove it with a picture. You can’t do it!

Line Postulate: There is exactly one line through any two points.

Postulate: Any line contains at least two points.

Postulate: The intersection of any two distinct lines will be a single point.

Plane Postulate: There is exactly one plane that contains any three non-collinear points.

Postulate: Any plane contains at least three non-collinear points.

Postulate: A line connecting points in a plane also lies within the plane.

Postulate: The intersection of two planes is a line.

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