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  Chapter 1: Basics of Geometry

  Points, Lines, and Planes

  Learning Objectives

  Understand the undefined terms point, line, and plane.

  Understand defined terms, including space, segment, and ray.

  Identify and apply basic postulates of points, lines, and planes.

  Draw and label terms in a diagram.

  Introduction

  Welcome to the exciting world of geometry! Ahead of you lie many exciting discoveries that will help you learn more about the world. Geometry is used in many areas—from art to science. For example, geometry plays a key role in construction, fashion design, architecture, and computer graphics. This course focuses on the main ideas of geometry that are the foundation of applications of geometry used everywhere. In this chapter, you’ll study the basic elements of geometry. Later you will prove things about geometric shapes using the vocabulary and ideas in this chapter—so make sure that you completely understand each of the concepts presented here before moving on.

  Undefined Terms

  The three basic building blocks of geometry are points, lines, and planes. These are undefined terms. While we cannot define these terms precisely, we can get an idea of what they are by looking at examples and models.

  A point is a location that has no size. To imagine a point, look at the period at the end of this sentence. Now imagine that period getting smaller and smaller until it disappears. A point describes a location, such as the location of the period, but a point has no size. We use dots (like periods) to represent points, but since the dots themselves occupy space, these dots are not points—we only use dots as representations. Points are labeled with a capital letter, as shown below.

  A line is an infinite series of points in a row. A line does not occupy space, so to imagine a line you can imagine the thinnest string you can think of, and shrink it until it occupies no space at all. A line has direction and location, but still does not take up space. Lines are sometimes referred to by one italicized letter, but they can also be identified by two points that are on the line. Lines are called one-dimensional, since they have direction in one dimension.

  The last undefined term is plane. You can think of a plane as a huge sheet of paper—so big that it goes on forever! Imagine the paper as thin as possible, and extend it up, down, left, and right. Planes can be named by letter, or by three points that lie in the plane. You already know one plane from your algebra class—the coordinate plane. Planes are called two-dimensional, since any point on a plane can be described by two numbers, called coordinates, as you learned in algebra.

  Notation Notes: As new terms are introduced, notation notes will help you learn how to write and say them.

  Points are named using a single capital letter. The first image shows points , , and .

  In the image of a line, the same line has several names. It can be called "line ," , or . The order of the letters does not matter when naming a line, so the same line can have many names. When using two points to name a line, you must use the line symbol above the letters.

  Planes are named using a script (cursive) letter or by naming three points contained in the plane. The illustrated plane can be called plane or “the plane defined by points , , and .”

  Example 1

  Which term best describes how San Diego, California, would be represented on a globe?

  A. point

  B. line

  C. plane

  A city is usually labeled with a dot, or point, on a globe. Though the city of San Diego occupies space, it is reduced when placed on the globe. Its label is merely to show a location with reference to the other cities, states, and countries on a globe. So, the correct answer is A.

  Example 2

  Which geometric object best models the surface of a movie screen?

  A. point

  B. line

  C. plane

  Airscreen auf dem James Dean Festival in Marion, USA

  The surface of a movie screen extends in two dimensions: up and down and left to right. This description most closely resembles a plane. So, the correct answer is C. Note that a plane is a model of the movie screen, but the screen is not actually a plane. In geometry, planes extend infinitely, but the movie screen does not.

  Defined Terms

  Now we can use point, line, and plane to define new terms. One word that has already been used is space. Space is the set of all points expanding 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. In algebra, the plane is adapted to model space by adding a third axis coming out of the page. The image below shows three perpendicular axes.

  Points are said to be collinear if they lie along the same line. The picture below shows points , , and are collinear. Point is non-collinear with the other three since it does not lie in the same line.

  Similarly, points and lines can be coplanar if they lie within the same plane. The diagram below shows two lines ( and ) and one point that are coplanar. It also shows line and point that are non-coplanar with and .

  A segment designates a portion of a line that has two endpoints. Segments are named by their endpoints.

  Notation Notes: Just like lines, segments are written with two capital letters. For segments we use a bar on top with no arrows. Segments can also be named in any order, so the segment above could be named or .

  A ray is a portion of a line that has only one endpoint and extends infinitely in the other direction. Rays are named by their endpoints and another point on the line. The endpoint always comes first in the name of a ray.

  Like segments, rays are named with two capital letters, and the symbol on top has one arrow. The ray is always named with the endpoint first, so we would write for the figure above.

  An intersection is the point or set of points where lines, planes, segments, or rays cross each other. Intersections are very important since you can study the different regions they create.

  In the image above, is the point of intersection of and . is the intersection of and .

  Example 3

  Which geometric object best models a straight road connecting two cities?

  A. ray

  B. line

  C. segment

  D. plane

  Since the straight road connects two distinct points (cities), and we are interested in the section between those two endpoints, the best term is segment. A segment has two endpoints. So, the correct answer is C.

  Example 4

  Which term best describes the relationship among the strings on a tennis racket?

  Photograph of a tennis racket and two balls

  A. collinear

  B. coplanar

  C. non-collinear

  D. non-coplanar

  The strings of a tennis racket are like intersecting segments. They also are all located on the plane made by the head of the racket. So, the best answer is B. Note that the strings are not really the same as segments and they are not exactly coplanar, but we can still use the geometric model of a plane for the head of a tennis racket, even if the model is not perfect.

  Basic Postulates

  Now that we have some basic vocabulary, we can talk about the rules of geo
metry. 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 cannot be proven.

  A theorem is a statement that can be proven true using postulates, definitions, logic, and other theorems we’ve already proven. Theorems are the “results” that are true given postulates and definitions. This section introduces a few basic postulates that you must understand as you move on to learn other theorems.

  The first of five postulates you will study in this lesson states that there is exactly one line through any two points. You could test this postulate easily with a ruler, a piece of paper, and a pencil. Use your pencil to draw two points anywhere on the piece of paper. Use your ruler to connect these two points. You’ll find that there is only one possible straight line that goes through them.

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

  Similarly, there is exactly one plane that contains any three non-collinear points. To illustrate this, ask three friends to hold up the tips of their pencils, and try and lay a piece of paper on top of them. If your friends line up their pencils (making the points collinear), there are an infinite number of possible planes. If one hand moves out of line, however, there is only one plane that will contain all three points. The following image shows five planes passing through three collinear points.

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

  If two coplanar points form a line, that line is also within the same plane.

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

  Sometimes lines intersect and sometimes they do not. When two lines do intersect, the intersection will be a single point. This postulate will be especially important when looking at angles and relationships between lines. As an extension of this, the final postulate for this lesson states that when two planes intersect they meet in a single line. The following diagrams show these relationships.

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

  Postulate: The intersection of two planes is a line.

  Example 5

  How many non-collinear points are required to identify a plane?

  A.

  B.

  C.

  D.

  The second postulate listed in this lesson states that you can identify a plane with three non-collinear points. It is important to label them as non-collinear points since there are infinitely many planes that contain collinear points. The answer is C.

  Example 6

  What geometric figure represents the intersection of the two planes below?

  A. point

  B. line

  C. ray

  D. plane

  The fifth postulate presented in this lesson says that the intersection of two planes is a line. This makes sense from the diagram as well. It is a series of points that extends infinitely in both directions, so it is definitely a line. The answer is B.

  Drawing and Labeling

  It is important as you continue your study of geometry to practice drawing geometric shapes. When you make geometric drawings, you need to be sure to follow the conventions of geometry so other people can “read” your drawing. For example, if you draw a line, be sure to include arrows at both ends. With only one arrow, it will appear as a ray, and without any arrows, people will assume that it is a line segment. Make sure you label your points, lines, and planes clearly, and refer to them by name when writing explanations. You will have many opportunities to hone your drawing skills throughout this geometry course.

  Example 7

  Draw and label the intersection of line and ray at point .

  To begin making this drawing, make a line with two points on it. Label the points and .

  Next, add the ray. The ray will have an endpoint and another point . The description says that the ray and line will intersect at , so point should be on . It is not important from this description in what direction points.

  The diagram above satisfies the conditions in the problem.

  Lesson Summary

  In this lesson, we explored points, lines, and planes. Specifically, we have learned:

  The significance of the undefined terms point, line, and plane.

  The significance of defined terms including space, segment, and ray.

  How to identify and apply basic postulates of points, lines, and planes.

  How to draw and label the terms you have studied in a diagram.

  These skills are the building blocks of geometry. It is important to have these concepts solidified in your mind as you explore other topics of geometry and mathematics.

  Points to Consider

  You can think of postulates as the basic rules of geometry. Other activities also have basic rules. For example, in the game of soccer one of the basic rules is that players are not allowed to use their hands to move the ball. How do the rules shape the way that the game is played? As you become more familiar with the geometric postulates, think about how the basic “rules of the game” in geometry determine what you can and cannot do.

  Now that you know some of the basics, we are going to look at how measurement is used in geometry.

  Review Questions

  Draw an image showing all of the following:

  intersecting

  Plane containing but not

  Name this line in three ways.

  What is the best possible geometric model for a soccer field? (See figure of soccer field.) Explain your answer.

  What type of geometric object is the intersection of a line and a plane? Draw your answer.

  What type of geometric object is made by the intersection of three planes? Draw your answer.

  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 this picture in as much detail as possible.

  True or false: Any two distinct points are collinear. Justify your answer.

  True or false: Any three distinct points determine a plane (or in other words, there is exactly one plane passing through any three points). Justify your answer.

  One of the statements in 8 or 9 is false. Rewrite the false statement to make it true.

  Review Answers

  Answers will vary, one possible example:

  ,, (and other answers are possible).

  A soccer field is like a plane since it is a flat two-dimensional surface.

  A line and a plane intersect at a point. See the diagram for answer 1 for an illustration. If were extended to be a line, then the intersection of and plane would be point .

  Three planes intersect at one point.

  A circle.

  intersects at point .

  True: The Line Postulate implies that you can always draw a line between any two points, so they must be collinear.

  False. Three collinear points could be at the intersection of an infinite number of planes. See the images of intersecting planes for an illustration of this.

  For 9 to be true, it should read: “Any three non-collinear points determine a plane.”

  Segments and Distance

  Learning Objectives

  Measure distances using different tools.

  Understand and apply the ruler postulate to measurement.

  Understand and apply the segment addition postulate to measurement.

  Use endpoints to identify distances on a coordinate grid.

  Introduction

  You have been using measurement for most of your life to understand quantities like weight, time, distance, area, and volume. Any time you have cooked a meal, bought something, or played a sport, measurement has played an important role. This lesson explores the postulates about measurement in geometry.

  Measuring Distances

  There are many different ways to identify measurements. This lesson will present some tha
t may be familiar, and probably a few that are new to you. Before we begin to examine distances, however, it is important to identify the meaning of distance in the context of geometry. The distance between two points is defined by the length of the line segment that connects them.

  The most common way to measure distance is with a ruler. Also, distance can be estimated using scale on a map. Practice this skill in the example below.

  Notation Notes: When we name a segment we use the endpoints and and overbar with no arrows. For example, "Segment " is written . The length of a segment is named by giving the endpoints without using an overline. For example, the length of is written . In some books you may also see , which means the same as , that is, it is the length of the segment with endpoints and .

  Example 1

  Use the scale to estimate the distance between Aaron’s house and Bijal’s house. Assume that the first third of the scale in black represents one inch.

  You need to find the distance between the two houses in the map. The scale shows a sample distance. Use the scale to estimate the distance. You will find that approximately three segments the length of the scale fit between the two points. Be careful—three is not the answer to this problem! As the scale shows one unit equal to two miles, you must multiply three units by two miles.