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 $xy-$ 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 $A$ , $M$ , and $P$ .
In the image of a line, the same line has several names. It can be called "line $g$ ," $\overleftrightarrow{PQ}$ , or $\overleftrightarrow{QP}$ . 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 $\leftrightarrow$ 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 $\mathit{M}$ or “the plane defined by points $A$ , $B$ , and $C$ .”

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 $x-y$ 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 $F$ , $G$ , and $H$ are collinear. Point $J$ 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 ( $\overleftrightarrow{RS}$ and $\overleftrightarrow{TV}$ ) and one point $(Q)$ that are coplanar. It also shows line $\overleftrightarrow{WX}$ and point $Z$ that are non-coplanar with $\overleftrightarrow{RS}$ and $Q$ .

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 $\overline{EF}$ or $\overline{FE}$ .

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 $\overrightarrow {CD}$ 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, $R$ is the point of intersection of $\overrightarrow{QR}$ and $\overrightarrow{SR}$ . $T$ is the intersection of $\overleftrightarrow{MN}$ and $\overleftrightarrow{PO}$ .

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 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 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. $1$

B. $2$

C. $3$

D. $4$

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 $\overleftrightarrow{AB}$ and ray $\overrightarrow{CD}$ at point $C$ .

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

Next, add the ray. The ray will have an endpoint $C$ and another point $D$ . The description says that the ray and line will intersect at $C$ , so point $C$ should be on $\overleftrightarrow{AB}$ . It is not important from this description in what direction $\overrightarrow{CD}$ 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:
1. $\overline{AB}$
2. $\overrightarrow {CD}$ intersecting $\overline{AB}$
3. Plane $P$ containing $\overline{AB}$ but not $\overrightarrow {CD}$
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:
$\overleftrightarrow{WX}$ , $\overleftrightarrow{YW}$ , $m$ (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 $\overrightarrow{CD}$ were extended to be a line, then the intersection of $\overrightarrow{CD}$ and plane $P$ would be point $C$ .
Three planes intersect at one point.
A circle.
$\overrightarrow{PQ}$ intersects $\overleftrightarrow{RS}$ at point $Q$ .
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 that 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 $AB$ " is written $\overline{AB}$ . The length of a segment is named by giving the endpoints without using an overline. For example, the length of $\overline{AB}$ is written $AB$ . In some books you may also see $m \overline{AB}$ , which means the same as $AB$ , that is, it is the length of the segment with endpoints $A$ and $B$ .

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.

$\mbox{3 units} \times \frac{\mbox{2 miles}} {\mbox{1 unit}} = \mbox{6 miles}$

The distance between the houses is about six miles.

You can also use estimation to identify measurements in other geometric figures. Remember to include words like approximately, about, or estimation whenever you are finding an estimated answer.

Ruler Postulate

You have probably been using rulers to measure distances for a long time and you know that a ruler is a tool with measurement markings.

Ruler Postulate: If you use a ruler to find the distance between two points, the distance will be the absolute value of the difference between the numbers shown on the ruler.

The ruler postulate implies that you do not need to start measuring at the zero mark, as long as you use subtraction to find the distance. Note, we say “absolute value” here since distances in geometry must always be positive, and subtraction can yield a negative result.

Example 2

What distance is marked on the ruler in the diagram below? Assume that the scale is marked in centimeters.

The way to use the ruler is to find the absolute value of difference between the numbers shown. The line segment spans from $3 \;\mathrm{cm}$ to $8 \;\mathrm{cm}$ .

$|3 - 8| = |-5| = 5$

The absolute value of the difference between the two numbers shown on the ruler is $5 \;\mathrm{cm}$ . So, the line segment is $5 \;\mathrm{cm}$ long.

Example 3

Use a ruler to find the length of the line segment below.

Line up the endpoints with numbers on your ruler and find the absolute value of the difference between those numbers. If you measure correctly, you will find that this segment measures $2.5\;\mathrm{inches}$ or $6.35\;\mathrm{centimeters}$ .

Segment Addition Postulate

Segment Addition Postulate: The measure of any line segment can be found by adding the measures of the smaller segments that comprise it.

That may seem like a lot of confusing words, but the logic is quite simple. In the diagram below, if you add the lengths of $\overline{AB}$ and $\overline{BC}$ , you will have found the length of $\overline{AC}$ . In symbols, $AB+BC=AC$ .

Use the segment addition postulate to put distances together.

Example 4

The map below shows the distances between three collinear towns. Assume that the first third of the scale in black represents one inch.

What is the distance between town $1$ and town $3$ ?

You can see that the distance between town $1$ and town $2$ is eight miles. You can also see that the distance between town $2$ and town $3$ is five miles. Using the segment addition postulate, you can add these values together to find the total distance between town $1$ and town $3$ .

$8+5=13$

The total distance between town $1$ and town $3$ is $13\;\mathrm{miles}$ .

Distances on a Grid

In algebra you most likely worked with graphing lines in the $x-y$ coordinate grid. Sometimes you can find the distance between points on a coordinate grid using the values of the coordinates. If the two points line up horizontally, look at the change of value in the $x-$ coordinates. If the two points line up vertically, look at the change of value in the $y-$ coordinates. The change in value will show the distance between the points. Remember to use absolute value, just like you did with the ruler. Later you will learn how to calculate distance between points that do not line up horizontally or vertically.

Example 5

What is the distance between the two points shown below?

The two points shown on the grid are at $(2,9)$ and $(2,3)$ . As these points line up vertically, look at the difference in the $y-$ values.

$|9-3|=|6|=6$

So, the distance between the two points is $6 \;\mathrm{units}$ .

Example 6

What is the distance between the two points shown below?

The two points shown on the grid are at $(-4, 4)$ and $(3, 4).$ These points line up horizontally, so look at the difference in the $x-$ values. Remember to take the absolute value of the difference between the values to find the distance.

$|(-4)-3|=|-7|=7$

The distance between the two points is $7\;\mathrm{units}$ .

Lesson Summary

In this lesson, we explored segments and distances. Specifically, we have learned:

How to measure distances using many different tools.
To understand and apply the Ruler Postulate to measurement.
To understand and apply the Segment Addition Postulate to measurement.
How to use endpoints to identify distances on a coordinate grid.

These skills are useful whenever performing measurements or calculations in diagrams. Make sure that you fully understand all concepts presented here before continuing in your study.

Review Questions

Use a ruler to measure the length of $\overline{AB}$ below.
According to the ruler in the following image, how long is the cockroach?
The ruler postulate says that we could have measured the cockroach in $2$ without using the $0 \;\mathrm{cm}$ marker as the starting point. If the same cockroach as the one in $2$ had its head at $6.5 \;\mathrm{cm}$ , where would its tail be on the ruler?
Suppose $M$ is exactly in the middle of $\overline{PQ}$ and $PM = 8 \;\mathrm{cm}$ . What is $PQ?$
What is $CE$ in the diagram below?
Find $x$ in the diagram below:
What is the length of the segment connecting $(-2,3)$ and $(-2, -7)$ in the coordinate plane? Justify your answer.
True or false: If $AB = 5\;\mathrm{cm}$ and $BC = 12\;\mathrm{cm}$ , then $AC = 17\;\mathrm{cm}$ .
True or false: $|a-b| = |b-a|$ .
One of the statements in 8 or 9 is false. Show why it is false, and then change the statement to make it true.

Review Answers

Answers will vary depending on scaling when printed and the units you use.
$4.5 \;\mathrm{cm}$ (yuck!).
The tail would be at either $11 \;\mathrm{cm}$ or $2 \;\mathrm{cm}$ , depending on which way the cockroach was facing.
$PQ = 2 \;\mathrm{(PM)} = 16 \;\mathrm{cm}$ .
$CE = 3 \;\mathrm{ft} + 9 \;\mathrm{ft} = 12 \;\mathrm{ft}$ .
$x = 36 \;\mathrm{km} - 7 \;\mathrm{km} = 29 \;\mathrm{km}$ .
Since the points are at the same $x-$ coordinate, we find the absolute value of the difference of the $y-$ coordinates. $|-7-3|=|-10|=10$
False.
True. $a - b = - (b - a)$ , but the absolute value sign makes them both positive.
Number 8 is false. See the diagram below for a counterexample. To make 8 true, we need to add something like: “If points $A$ , $B$ , and $C$ are collinear, and $B$ is between $A$ and $C$ , then if $AB = 5 \;\mathrm{cm}$ and $BC = 12 \;\mathrm{cm}$ , then $AC = 17 \;\mathrm{cm}$ .”

Rays and Angles

Learning Objectives

Understand and identify rays.
Understand and classify angles.
Understand and apply the protractor postulate.
Understand and apply the angle addition postulate.

Introduction

Now that you know about line segments and how to measure them, you can apply what you have learned to other geometric figures. This lesson deals with rays and angles, and you can apply much of what you have already learned. We will try to help you see the connections between the topics you study in this book instead of dealing with them in isolation. This will give you a more well-rounded understanding of geometry and make you a better problem solver.

Rays

A ray is a part of a line with exactly one endpoint that extends infinitely in one direction. Rays are named by their endpoint and a point on the ray.

The ray above is called $\overrightarrow{AB}$ . The first letter in the ray’s name is always the endpoint of the ray, it doesn’t matter which direction the ray points.

Rays can represent a number of different objects in the real world. For example, the beam of light extending from a flashlight that continues forever in one direction is a ray. The flashlight would be the endpoint of the ray, and the light continues as far as you can imagine so it is the infinitely long part of the ray. Are there other real-life objects that can be represented as rays?

Example 1

Which of the figures below shows $\overrightarrow{GH}?$

Remember that a ray has one endpoint and extends infinitely in one direction. Choice A is a line segment since it has two endpoints. Choice B has one endpoint and extends infinitely in one direction, so it is a ray. Choice C has no endpoints and extends infinitely in two directions — it is a line. Choice D also shows a ray with endpoint $H.$ Since we need to identify $\overrightarrow{GH}$ with endpoint $G$ , we know that choice B is correct.

Example 2

Use this space to draw $\overrightarrow{RT}$ .

Remember that you are not expected to be an artist. In geometry, you simply need to draw figures that accurately represent the terms in question. This problem asks you to draw a ray. Begin with a line segment. Use your ruler to draw a straight line segment of any length.

Now draw an endpoint on one end and an arrow on the other.

Finally, label the endpoint $R$ and another point on the ray $T$ .

The diagram above shows $\overrightarrow{RT}$ .

Angles

An angle is formed when two rays share a common endpoint. That common endpoint is called the vertex and the two rays are called the sides of the angle. In the diagram below, $\overrightarrow{AB}$ and $\overrightarrow{AT}$ form an angle, $\angle BAT$ , or $\angle A$ for short. The symbol $\angle$ is used for naming angles.

The same basic definition for angle also holds when lines, segments, or rays intersect.

Notation Notes:

Angles can be named by a number, a single letter at the vertex, or by the three points that form the angle. When an angle is named with three letters, the middle letter will always be the vertex of the angle. In the diagram above, the angle can be written $\angle{BAT}$ , or $\angle{TAB}$ , or $\angle{A}$ . You can use one letter to name this angle since point $A$ is the vertex and there is only one angle at point $A$ .
If two or more angles share the same vertex, you MUST use three letters to name the angle. For example, in the image below it is unclear which angle is referred to by $\angle{L}$ . To talk about the angle with one arc, you would write $\angle{KLJ}$ . For the angle with two arcs, you’d write $\angle{JLM}$ .

We use a ruler to measure segments by their length. But how do we measure an angle? The length of the sides does not change how wide an angle is “open.” Instead of using length, the size of an angle is measured by the amount of rotation from one side to another. By definition, a full turn is defined as $360$ degrees. Use the symbol $^\circ$ for degrees. You may have heard “ $360$ ” used as slang for a “full circle” turn, and this expression comes from the fact that a full rotation is $360^\circ$ .

The angle that is made by rotating through one-fourth of a full turn is very special. It measures $\frac{1} {4} \times 360^\circ = 90^\circ$ and we call this a right angle. Right angles are easy to identify, as they look like the corners of most buildings, or a corner of a piece of paper.

A right angle measures exactly $90^\circ$ .

Right angles are usually marked with a small square. When two lines, two segments, or two rays intersect at a right angle, we say that they are perpendicular. The symbol $\bot$ is used for two perpendicular lines.

An acute angle measures between $0^\circ$ and $90^\circ$ .

An obtuse angle measures between $90^\circ$ and $180^\circ$ .

A straight angle measures exactly $180^\circ$ . These are easy to spot since they look like straight lines.

You can use this information to classify any angle you see.

Example 3

What is the name and classification of the angle below?

Begin by naming this angle. It has three points labeled and the vertex is $U$ . So, the angle will be named $\angle{TUV}$ or just $\angle{U}$ . For the classification, compare the angle to a right angle. $\angle{TUV}$ opens wider than a right angle, and less than a straight angle. So, it is obtuse.

Example 4

What term best describes the angle formed by Clinton and Reeve streets on the map below?

The intersecting streets form a right angle. It is a square corner, so it measures $90^\circ$ .

Protractor Postulate

In the last lesson, you studied the ruler postulate. In this lesson, we’ll explore the Protractor Postulate. As you can guess, it is similar to the ruler postulate, but applied to angles instead of line segments. A protractor is a half-circle measuring device with angle measures marked for each degree. You measure angles with a protractor by lining up the vertex of the angle on the center of the protractor and then using the protractor postulate (see below). Be careful though, most protractors have two sets of measurements—one opening clockwise and one opening counterclockwise. Make sure you use the same scale when reading the measures of the angle.

Protractor Postulate: For every angle there is a number between $0$ and $180$ that is the measure of the angle in degrees. You can use a protractor to measure an angle by aligning the center of the protractor on the vertex of the angle. The angle's measure is then the absolute value of the difference of the numbers shown on the protractor where the sides of the angle intersect the protractor.

It is probably easier to understand this postulate by looking at an example. The basic idea is that you do not need to start measuring an angle at the zero mark, as long as you find the absolute value of the difference of the two measurements. Of course, starting with one side at zero is usually easier. Examples 5 and 6 show how to use a protractor to measure angles.

Notation Note: When we talk about the measure of an angle, we use the symbols $m\angle$ . So for example, if we used a protractor to measure $\angle{TUV}$ in example 3 and we found that it measured $120^\circ$ , we could write $m \angle{TUV} = 120^\circ$ .

Example 5

What is the measure of the angle shown below?

This angle is lined up with a protractor at $0^\circ$ , so you can simply read the final number on the protractor itself. Remember you can check that you are using the correct scale by making sure your answer fits your angle. If the angle is acute, the measure of the angle should be less than $90^\circ$ . If it is obtuse, the measure will be greater than $90^\circ$ . In this case, the angle is acute, so its measure is $50^\circ$ .

Example 6

What is the measure of the angle shown below?

This angle is not lined up with the zero mark on the protractor, so you will have to use subtraction to find its measure.

Using the inner scale, we get $|140-15|=|125|=125^{\circ}$ .

Using the outer scale, $|40-165|=|-125|=125^{\circ}$ .

Notice that it does not matter which scale you use. The measure of the angle is $125^\circ$ .

Example 7

Use a protractor to measure $\angle{RST}$ below.

You can either line it up with zero, or line it up with another number and find the absolute value of the differences of the angle measures at the endpoints. Either way, the result is $100^\circ$ . The angle measures $100^\circ$ .

Multimedia Link The following applet gives you practice measuring angles with a protractor Measuring Angles Applet.

Angle Addition Postulate

You have already encountered the ruler postulate and the protractor postulate. There is also a postulate about angles that is similar to the Segment Addition Postulate.

Angle Addition Postulate: The measure of any angle can be found by adding the measures of the smaller angles that comprise it. In the diagram below, if you add $m\angle{ABC}$ and $m\angle{CBD}$ , you will have found $m\angle{ABD}$ .

Use this postulate just as you did the segment addition postulate to identify the way different angles combine.

Example 8

What is $m\angle{QRT}$ in the diagram below?

You can see that $m\angle{QRS}$ is $15^\circ$ . You can also see that $m\angle{SRT}$ is $30^\circ$ . Using the angle addition postulate, you can add these values together to find the total $m\angle{QRT}$ .

$15+30=45$

So, $m\angle{QRT}$ is $45^\circ$ .

Example 9

What is $m\angle{LMN}$ in the diagram below given $m\angle{LMO} = 85^\circ$ and $m\angle{NMO} = 53^\circ$ ?

To find $m\angle{LMN}$ , you must subtract $m\angle{NMO}$ from $m\angle{LMO}$ .

$85-53=32$

So, $m\angle{LMN} = 32^{\circ}$ .

Lesson Summary

In this lesson, we explored rays and angles. Specifically, we have learned:

To understand and identify rays.
To understand and classify angles.
To understand and apply the Protractor Postulate.
To understand and apply the Angle Addition Postulate.

These skills are useful whenever studying rays and angles. Make sure that you fully understand all concepts presented here before continuing in your study.

Review Questions

Use this diagram for questions 1-4.

Give two possible names for the ray in the diagram.
Give four possible names for the line in the diagram.
Name an acute angle in the diagram.
Name an obtuse angle in the diagram.
Name a straight angle in the diagram.
Which angle can be named using only one letter?
Explain why it is okay to name some angles with only one angle, but with other angles this is not okay.
Use a protractor to find $m\angle{PQR}$ below:
Given $m \angle{FNI} = 125^\circ$ and $m \angle{HNI} = 50^\circ$ , find $m \angle{FNH}$ .
True or false: Adding two acute angles will result in an obtuse angle. If false, provide a counterexample.

Review Answers

$CD$ or $CE$
$BD$ , $DB$ , $AB$ , or $BA$ are four possible answers. There are more (how many?)
$BDC$
$BDE$ or $BCD$ or $CDA$
$BDA$
Angle $C$
If there is more than one angle at a given vertex, then you must use three letters to name the angle. If there is only one angle at a vertex (as in angle $C$ above) then it is permissible to name the angle with one letter.
$|(50-130)| = |(-80)| = 80.$
$m\angle{FNH} = |125-50| = |75|=75^{\circ}$ .
False. For a counterexample, suppose two acute angles measure $30^\circ$ and $45^\circ$ , then the sum of those angles is $75^\circ$ , but $75^\circ$ is still acute. See the diagram for a counterexample:

Segments and Angles

Learning Objectives

Understand and identify congruent line segments.
Identify the midpoint of line segments.
Identify the bisector of a line segment.
Understand and identify congruent angles.
Understand and apply the Angle Bisector Postulate.

Introduction

Now that you have a better understanding of segments, angles, rays, and other basic geometric shapes, we can study the ways in which they can be divided. Any time you come across a segment or an angle, there are different ways to separate it into parts.

Congruent Line Segments

One of the most important words in geometry is congruent. This term refers to geometric objects that have exactly the same size and shape. Two segments are congruent if they have the same length.

Notation Notes:

When two things are congruent we use the symbol $\cong$ . For example if $\overline{AB}$ is congruent to $\overline{CD}$ , then we would write $\overline{AB} \cong \overline{CD}$ .
When we draw congruent segments, we use tic marks to show that two segments are congruent.
If there are multiple pairs of congruent segments (which are not congruent to each other) in the same picture, use two tic marks for the second set of congruent segments, three for the third set, and so on. See the two following illustrations.

Recall that the length of segment $\overline{AB}$ can be written in two ways: $m\overline{AB}$ or simply $AB$ . This might be a little confusing at first, but it will make sense as you use this notation more and more. Let’s say we used a ruler and measured $\overline{AB}$ and we saw that it had a length of $5 \;\mathrm{cm}$ . Then we could write $m\overline{AB} = 5 \;\mathrm{cm}$ , or $AB = 5 \;\mathrm{cm}$ .

If we know that $\overline{AB} \cong \overline{CD}$ , then we can write $m\overline{AB} = m\overline{CD}$ or simply $AB = CD$ .

You can prove two segments are congruent in a number of ways. You can measure them to find their lengths using any units of measurement—the units do not matter as long as you use the same units for both measurements. Or, if the segments are drawn in the $x-y$ plane, you can also find their lengths on the coordinate grid. Later in the course you will learn other ways to prove two segments are congruent.

Example 1

Henrietta drew a line segment on a coordinate grid as shown below.

She wants to draw another segment congruent to the first that begins at $(-1,1)$ and travels straight up (that is, in the $+y$ direction). What will be the coordinates of its second endpoint?

You will have to solve this problem in stages. The first step is to identify the length of the segment drawn onto the grid. It begins at $(2,3)$ and ends at $(6,3)$ . So, its length is $4\;\mathrm{units}$ .

The next step is to draw the second segment. Use a pencil to create the segment according to the specifications in the problem. You know that the segment needs to be congruent to the first, so it will be $4\;\mathrm{units}$ long. The problem also states that it travels straight up from the point $(-1,1)$ . Draw in the point at $(-1,1)$ and make a line segment $4\;\mathrm{units}$ long that travels straight up.

Now that you have drawn in the new segment, use the grid to identify the new endpoint. It has an $x-$ coordinate of $-1$ and a $y-$ coordinate of $5$ . So, its coordinates are $(-1,5)$ .

Segment Midpoints

Now that you understand congruent segments, there are a number of new terms and types of figures you can explore. A segment midpoint is a point on a line segment that divides the segment into two congruent segments. So, each segment between the midpoint and an endpoint will have the same length. In the diagram below, point $B$ is the midpoint of segment $\overline{AC}$ since $\overline{AB}$ is congruent to $\overline{BC}$ .

There is even a special postulate dedicated to midpoints.

Segment Midpoint Postulate: Any line segment will have exactly one midpoint—no more, and no less.

Example 2

Nandi and Arshad measure and find that their houses are $10\;\mathrm{miles}$ apart. If they agree to meet at the midpoint between their two houses, how far will each of them travel?

The easiest way to find the distance to the midpoint of the imagined segment connecting their houses is to divide the length by $2$ .

$10 \div 2 = 5$

So, each person will travel five miles to meet at the midpoint between Nandi’s and Arshad’s houses.

Segment Bisectors

Now that you know how to find midpoints of line segments, you can explore segment bisectors. A bisector is a line, segment, or ray that passes through a midpoint of another segment. You probably know that the prefix “bi” means two (think about the two wheels of a bicycle). So, a bisector cuts a line segment into two congruent parts.

Example 3

Use a ruler to draw a bisector of the segment below.

The first step in identifying a bisector is finding the midpoint. Measure the line segment to find that it is $4 \;\mathrm{cm}$ long. To find the midpoint, divide this distance by $2$ .

$4 \div 2 = 2$

So, the midpoint will be $2 \;\mathrm{cm}$ from either endpoint on the segment. Measure $2 \;\mathrm{cm}$ from an endpoint and draw the midpoint.

To complete the problem, draw a line segment that passes through the midpoint. It doesn’t matter what angle this segment travels on. As long as it passes through the midpoint, it is a bisector.

Congruent Angles

You already know that congruent line segments have exactly the same length. You can also apply the concept of congruence to other geometric figures. When angles are congruent, they have exactly the same measure. They may point in different directions, have different side lengths, have different names or other attributes, but their measures will be equal.

Notation Notes:

When writing that two angles are congruent, we use the congruent symbol: $\angle{ABC} \cong \angle{ZYX}$ . Alternatively, the symbol $m\angle{ABC}$ refers to the measure of $\angle{ABC}$ , so we could write $m\angle{ABC} = m \angle{ZYX}$ and that has the same meaning as $\angle{ABC} \cong \angle{ZYX}$ . You may notice then, that numbers (such as measurements) are equal while objects (such as angles and segments) are congruent.
When drawing congruent angles, you use an arc in the middle of the angle to show that two angles are congruent. If two different pairs of angles are congruent, use one set of arcs for one pair, then two for the next pair and so on.

Use algebra to find a way to solve the problem below using this information.

Example 4

The two angles shown below are congruent.

What is the measure of each angle?

This problem combines issues of both algebra and geometry, so make sure you set up the problem correctly. It is given that the two angles are congruent, so they must have the same measurements. Therefore, you can set up an equation in which the expressions representing the angle measures are equal to each other.

$5x+7=3x+23$

Now that you have an equation with one variable, you can solve for the value of $x$ .

$5x+7 & = 3x+23\ 5x-3x & = 23-7\ 2x & = 16\ x & = 8$

So, the value of $x$ is $8$ . You are not done, however. Use this value of $x$ to find the measure of one of the angles in the problem.

$m\angle{ABC} &=5x+7\ &=5(8)+7\ &=40+7\ &=47$

Finally, we know $m\angle{ABC}=m\angle{XYZ}$ , so both of the angles measure $47^\circ$ .

Angle Bisectors

If a segment bisector divides a segment into two congruent parts, you can probably guess what an angle bisector is. An angle bisector divides an angle into two congruent angles, each having a measure exactly half of the original angle.

Angle Bisector Postulate: Every angle has exactly one bisector.

Example 5

The angle below measures $136^\circ$ .

If a bisector is drawn in this angle, what will be the measure of the new angles formed?

This is similar to the problem about the midpoint between the two houses. To find the measurements of the smaller angles once a bisector is drawn, divide the original angle measure by $2$ :

$136 \div 2 = 68$

So, each of the newly formed angles would measure $68^\circ$ when the $136^\circ$ angle is bisected.

Lesson Summary

In this lesson, we explored segments and angles. Specifically, we have learned:

How to understand and identify congruent line segments.
How to identify the midpoint of line segments.
How to identify the bisector of a line segment.
How to understand and identify congruent angles.
How to understand and apply the Angle Bisector Postulate.

These skills are useful whenever performing measurements or calculations in diagrams. Make sure that you fully understand all concepts presented here before continuing in your study.

Review Questions

Copy the figure below and label it with the following information:
1. $\angle{A} \cong \angle{C}$
2. $\angle{B} \cong \angle{D}$
3. $\overline{AB} \cong \overline{AD}$
Sketch and label an angle bisector $\overrightarrow{RU}$ of $\angle{SRT}$ below.
If we know that $m\angle{SRT} = 64^\circ$ , what is $m\angle{SRU}?$

Use the following diagram of rectangle $ACEF$ for questions 4-10. (For these problems you can assume that opposite sides of a rectangle are congruent—later you will prove this is true.)

Given that $H$ is the midpoint of $\overline{AE}$ and $\overline{DG}$ , find the following lengths:

$GH =$
$AB =$
$AC =$
$HE =$
$AE =$
$CE =$
$GF =$
How many copies of $\triangle ABH$ can fit inside rectangle $ACEF?$

Review Answers

$32^\circ$
$GH = 12 \;\mathrm{in}$
$AB = 12 \;\mathrm{in}$
$AC = 24 \;\mathrm{in}$
$HE = 12 \;\mathrm{in}$
$AE = 26 \;\mathrm{in}$
$CE = 10 \;\mathrm{in}$
$GF = 5 \;\mathrm{in}$
$8$

Angle Pairs

Learning Objectives

Understand and identify complementary angles.
Understand and identify supplementary angles.
Understand and utilize the Linear Pair Postulate.
Understand and identify vertical angles.

Introduction

In this lesson you will learn about special angle pairs and prove the vertical angles theorem, one of the most useful theorems in geometry.

Complementary Angles

A pair of angles are Complementary angles if the sum of their measures is $90^\circ$ .

Complementary angles do not have to be congruent to each other. Rather, their only defining quality is that the sum of their measures is equal to the measure of a right angle: $90^\circ$ . If the outer rays of two adjacent angles form a right angle, then the angles are complementary.

Example 1

The two angles below are complementary. $m\angle{GHI}=x$ . What is the value of $x$ ?

Since you know that the two angles must sum to $90^\circ$ , you can create an equation. Then solve for the variable. In this case, the variable is $x$ .

$34+x &= 90 \ 34 + x -4 & = 90-34\ x & =56$

Thus, the value of $x$ is $56^\circ$ .

Example 2

The two angles below are complementary. What is the measure of each angle?

This problem is a bit more complicated than the first example. However, the concepts are the same. If you add the two angles together, the sum will be $90^\circ$ . So, you can set up an algebraic equation with the values presented.

$(7r+6) + (8r+9) = 90$

The best way to solve this problem is to solve the equation above for $r$ . Then, you must substitute the value for $r$ back into the original expressions to find the value of each angle.

$(7r+6)+(8r+9) &= 90\ 15r + 15 &= 90 \ 15r + 15 - 15 &= 90-15 \ 15r &= 75 \ \frac{15r}{15} &= \frac{75}{15} \ r &= 5$

The value of $r$ is $5$ . Now substitute this value back into the expressions to find the measures of the two angles in the diagram.

$7r& + 6 & 8r& + 9\ 7(5)& + 6 & 8(5)& + 9\ 35& + 6 & 40& + 9\ 41& & 49&$

$m\angle{JKL}=41^{\circ}$ and $m\angle{GHI}=49^{\circ}$ . You can check to make sure these numbers are accurate by verifying if they are complementary.

$41 + 49 = 90$

Since these two angle measures sum to $90^\circ$ , they are complementary.

Supplementary Angles

Two angles are supplementary if their measures sum to $180^\circ$ .

Just like complementary angles, supplementary angles need not be congruent, or even touching. Their defining quality is that when their measures are added together, the sum is $180^\circ$ . You can use this information just as you did with complementary angles to solve different types of problems.

Example 3

The two angles below are supplementary. If $m\angle{MNO}=78^{\circ}$ , what is $m\angle{PQR}$ ?

This process is very straightforward. Since you know that the two angles must sum to $180^\circ$ , you can create an equation. Use a variable for the unknown angle measure and then solve for the variable. In this case, let's substitute $y$ for $m\angle{PQR}$ .

$78+y&=180\ 78+y-78&=180-78\ y &=102$

So, the measure of $y=102$ and thus $m\angle{PQR}=102^{\circ}$ .

Example 4

What is the measure of two congruent, supplementary angles?

There is no diagram to help you visualize this scenario, so you’ll have to imagine the angles (or even better, draw it yourself by translating the words into a picture!). Two supplementary angles must sum to $180^\circ$ . Congruent angles must have the same measure. So, you need to find two congruent angles that are supplementary. You can divide $180^\circ$ by two to find the value of each angle.

$180 \div 2 = 90$

Each congruent, supplementary angle will measure $90^\circ$ . In other words, they will be right angles.

Linear Pairs

Before we talk about a special pair of angles called linear pairs, we need to define adjacent angles. Two angles are adjacent if they share the same vertex and one side, but they do not overlap. In the diagram below, $\angle{PQR}$ and $\angle{RQS}$ are adjacent.

However, $\angle{PQR}$ and $\angle{PQS}$ are not adjacent since they overlap (i.e. they share common points in the interior of the angle).

Now we are ready to talk about linear pairs. A linear pair is two angles that are adjacent and whose non-common sides form a straight line. In the diagram below, $\angle{MNP}$ and $\angle{PNO}$ are a linear pair. Note that $\overleftrightarrow{MO}$ is a line.

Linear pairs are so important in geometry that they have their own postulate.

Linear Pair Postulate: If two angles are a linear pair, then they are supplementary.

Example 5

The two angles below form a linear pair. What is the value of each angle?

If you add the two angles, the sum will be $180^\circ$ . So, you can set up an algebraic equation with the values presented.

$(3q)+(15q+18)=180$

The best way to solve this problem is to solve the equation above for $q$ . Then, you must plug the value for $q$ back into the original expressions to find the value of each angle.

$(3q) + (15q + 18) & = 180\ 18q+18 &=180\ 18q &= 180-18\ 18q &=162\ \frac{18q}{18} &=\frac{162}{18}\ q &=9$

The value of $q$ is $9$ . Now substitute this value back into the expressions to determine the measures of the two angles in the diagram.

$3q& & 15q& + 18\ 3(9)& & 15(9)& + 18\ 27& & 135& + 18\ & & 153&$

The two angles in the diagram measure $27^\circ$ and $153^\circ$ . You can check to make sure these numbers are accurate by verifying if they are supplementary.

$27+153=180$

Vertical Angles

Now that you understand supplementary and complementary angles, you can examine more complicated situations. Special angle relationships are formed when two lines intersect, and you can use your knowledge of linear pairs of angles to explore each angle further.

Vertical angles are defined as two non-adjacent angles formed by intersecting lines. In the diagram below, $\angle{1}$ and $\angle{3}$ are vertical angles. Also, $\angle{4}$ and $\angle{2}$ are vertical angles.

Suppose that you know $m\angle{1}=100^{\circ}$ . You can use that information to find the measurement of all the other angles. For example, $\angle{1}$ and $\angle{2}$ must be supplementary since they are a linear pair. So, to find $m\angle{2}$ , subtract $100^\circ$ from $180^\circ$ .

$m\angle{1}+m\angle{2} &=180\ 100+ m\angle{2} &= 180\ m\angle{2}& = 180-100\ m\angle{2}& = 80$

So $\angle{2}$ measures $80^\circ$ . Knowing that angles $2$ and $3$ are also supplementary means that $m\angle{3}=100^{\circ}$ , since the sum of $100^\circ$ and $80^\circ$ is $180^\circ$ . If angle $3$ measures $100^\circ$ , then the measure of angle $4$ must be $80^\circ$ , since $3$ and $4$ are also supplementary. Notice that angles $1$ and $3$ are congruent $(100^\circ)$ and $2$ and $4$ are congruent $(80^\circ)$ .

The Vertical Angles Theorem states that if two angles are vertical angles then they are congruent.

We can prove the vertical angles theorem using a process just like the one we used above. There was nothing special about the given measure of $\angle{1}$ . Here is proof that vertical angles will always be congruent: Since $\angle{1}$ and $\angle{2}$ form a linear pair, we know that they are supplementary: $m\angle{1}+m\angle{2}=180^{\circ}$ . For the same reason, $\angle{2}$ and $\angle{3}$ are supplementary: $m\angle{2}+m\angle{3}=180^{\circ}$ . Using a substitution, we can write $m\angle{1}+m\angle{2}=m\angle{2}+m\angle{3}$ . Finally, subtracting $m\angle{2}$ on both sides yields $m\angle{1}=m\angle{3}$ . Or, by the definition of congruent angles, $\angle{1} \cong \angle{3}$ .

Use your knowledge of vertical angles to solve the following problem.

Example 6

What is $m\angle{STU}$ in the diagram below?

Using your knowledge of intersecting lines, you can identify that $\angle{STU}$ is vertical to the angle marked $18^\circ$ . Since vertical angles are congruent, they will have the same measure. So, $m \angle{STU}$ is also equal to $18^\circ$ .

Lesson Summary

In this lesson, we explored angle pairs. Specifically, we have learned:

How to understand and identify complementary angles.
How to understand and identify supplementary angles.
How to understand and utilize the Linear Pair Postulate.
How to understand and identify vertical angles.

The relationships between different angles are used in almost every type of geometric application. Make sure that these concepts are retained as you progress in your studies.

Review Questions

Find the measure of the angle complementary to if
1. $45^\circ$
2. $82^\circ$
3. $19^\circ$
4. $z^\circ$
Find the measure of the angle supplementary to if
1. $45^\circ$
2. $118^\circ$
3. $32^\circ$
4. $x^\circ$
Find $m\angle{ABD}$ and $m\angle{DBC}$ .
Given $m\angle{EFG} = 20^\circ$ , Find $m\angle{HFG}$ .

Use the diagram below for exercises 5 and 6. Note that $\overline{NK} \perp$ $\overleftrightarrow{IL}$ .
Identify each of the following (there may be more than one correct answer for some of these questions).
1. Name one pair of vertical angles.
2. Name one linear pair of angles.
3. Name two complementary angles.
4. Nam two supplementary angles.
Given that , find
1. $m\angle{JNK}$ .
2. $m\angle{KNL}$ .
3. $m\angle{MNL}$ .
4. $m\angle{MNI}$ .

Review Answers

1. $45^\circ$
2. $8^\circ$
3. $81^\circ$
4. $(90 - z)^\circ$
1. $135^\circ$
2. $62^\circ$
3. $148^\circ$
4. $(180 - x)^\circ$
$m \angle{ABD} = 73^\circ$ , $m \angle{DBC} = 107^\circ$
$m \angle{HFG} = 70^\circ$
1. $\angle{JNI}$ and $\angle{MNL}$ (or $\angle{INM}$ and $\angle{JNL}$ also works);
2. $\angle{INM}$ and $\angle{MNL}$ (or $\angle{INK}$ and $\angle{KNL}$ also works);
3. $\angle{INK}$ and $\angle{JNK}$ ;
4. same as (b) $\angle{INM}$ and $\angle{MNL}$ (or $\angle{INK}$ and $\angle{KNL}$ also works).
1. $27^\circ$
2. $90^\circ$
3. $63^\circ$
4. $117^\circ$

Classifying Triangles

Learning Objectives

Define triangles.
Classify triangles as acute, right, obtuse, or equiangular.
Classify triangles as scalene, isosceles, or equilateral.

Introduction

By this point, you should be able to readily identify many different types of geometric objects. You have learned about lines, segments, rays, planes, as well as basic relationships between many of these figures. Everything you have learned up to this point is necessary to explore the classifications and properties of different types of shapes. The next two sections focus on two-dimensional shapes—shapes that lie in one plane. As you learn about polygons, use what you know about measurement and angle relationships in these sections.

Defining Triangles

The first shape to examine is the triangle. Though you have probably heard of triangles before, it is helpful to review the formal definition. A triangle is any closed figure made by three line segments intersecting at their endpoints. Every triangle has three vertices (points at which the segments meet), three sides (the segments themselves), and three interior angles (formed at each vertex). All of the following shapes are triangles.

You may have learned in the past that the sum of the interior angles in a triangle is always $180^\circ$ . Later we will prove this property, but for now you can use this fact to find missing angles. Other important properties of triangles will be explored in later chapters.

Example 1

Which of the figures below are not triangles?

To solve this problem, you must carefully analyze the four shapes in the answer choices. Remember that a triangle has three sides, three vertices, and three interior angles. Choice A fits this description, so it is a triangle. Choice B has one curved side, so its sides are not exclusively line segments. Choice C is also a triangle. Choice D, however, is not a closed shape. Therefore, it is not a triangle. Choices B and D are not triangles.

Example 2

How many triangles are in the diagram below?

To solve this problem, you must carefully count the triangles of different size. Begin with the smallest triangles. There are $16$ small triangles.

Now count the triangles that are formed by four of the smaller triangles, like the one below.

There are a total of seven triangles of this size, if you remember to count the inverted one in the center of the diagram.

Next, count the triangles that are formed by nine of the smaller triangles. There are three of these triangles. And finally, there is one triangle formed by $16$ smaller triangles.

Now, add these numbers together.

$16 + 7 + 3 + 1 = 27$

So, there are a total of $27$ triangles in the figure shown.

Classifications by Angles

Earlier in this chapter, you learned how to classify angles as acute, obtuse, or right. Now that you know how to identify triangles, we can separate them into classifications as well. One way to classify a triangle is by the measure of its angles. In any triangle, two of the angles will always be acute. This is necessary to keep the total sum of the interior angles at $180^\circ$ . The third angle, however, can be acute, obtuse, or right.

This is how triangles are classified. If a triangle has one right angle, it is called a right triangle.

If a triangle has one obtuse angle, it is called an obtuse triangle.

If all of the angles are acute, it is called an acute triangle.

The last type of triangle classifications by angles occurs when all angles are congruent. This triangle is called an equiangular triangle.

Example 3

Which term best describes $\triangle{RST}$ below?

The triangle in the diagram has two acute angles. But, $m\angle{RST}=92^{\circ}$ so $\angle{RST}$ is an obtuse angle. If the angle measure were not given, you could check this using the corner of a piece of notebook paper or by measuring the angle with a protractor. An obtuse angle will be greater than $90^\circ$ (the square corner of a paper) and less than $180^\circ$ (a straight line). Since one angle in the triangle above is obtuse, it is an obtuse triangle.

Classifying by Side Lengths

There are more types of triangle classes that are not based on angle measure. Instead, these classifications have to do with the sides of the triangle and their relationships to each other. When a triangle has all sides of different length, it is called a scalene triangle.

When at least two sides of a triangle are congruent, the triangle is said to be an isosceles triangle.

Finally, when a triangle has sides that are all congruent, it is called an equilateral triangle. Note that by the definitions, an equilateral triangle is also an isosceles triangle.

Example 4

Which term best describes the triangle below?

A. scalene

B. isosceles

C. equilateral

To classify the triangle by side lengths, you have to examine the relationships between the sides. Two of the sides in this triangle are congruent, so it is an isosceles triangle. The correct answer is B.

Lesson Summary

In this lesson, we explored triangles and their classifications. Specifically, we have learned:

How to define triangles.
How to classify triangles as acute, right, obtuse, or equiangular.
How to classify triangles as scalene, isosceles, or equilateral.

These terms or concepts are important in many different types of geometric practice. It is important to have these concepts solidified in your mind as you explore other topics of geometry and mathematics.

Review Questions

Exercises 1-5: Classify each triangle by its sides and by its angles. If you do not have enough information to make a classification, write “not enough information.”

Sketch an equiangular triangle. What must be true about the sides?
Sketch an obtuse isosceles triangle.
True or false: A right triangle can be scalene.
True or false: An obtuse triangle can have more than one obtuse angle.
One of the answers in 8 or 9 is false. Sketch an illustration to show why it is false, and change the false statement to make it true.

Review Answers

A is an acute scalene triangle.
B is an equilateral triangle.
C is a right isosceles triangle.
D is a scalene triangle. Since we don’t know anything about the angles, we cannot assume it is a right triangle, even though one of the angles looks like it may be $90^\circ$ .
E is an obtuse scalene triangle.
If a triangle is equiangular then it is also equilateral, so the sides are all congruent.
Sketch below:
True.
False.
9 is false since the three sides would not make a triangle. To make the statement true, it should say: “An obtuse triangle has exactly one obtuse angle.”

Classifying Polygons

Learning Objectives

Define polygons.
Understand the difference between convex and concave polygons.
Classify polygons by number of sides.
Use the distance formula to find side lengths on a coordinate grid.

Introduction

As you progress in your studies of geometry, you can examine different types of shapes. In the last lesson, you studied the triangle, and different ways to classify triangles. This lesson presents other shapes, called polygons. There are many different ways to classify and analyze these shapes. Practice these classification procedures frequently and they will get easier and easier.

Defining Polygons

Now that you know what a triangle is, you can learn about other types of shapes. Triangles belong to a larger group of shapes called polygons. A polygon is any closed planar figure that is made entirely of line segments that intersect at their endpoints. Polygons can have any number of sides and angles, but the sides can never be curved.

The segments are called the sides of the polygons, and the points where the segments intersect are called vertices. Note that the singular of vertices is vertex.

The easiest way to identify a polygon is to look for a closed figure with no curved sides. If there is any curvature in a shape, it cannot be a polygon. Also, the points of a polygon must all lie within the same plane (or it wouldn’t be two-dimensional).

Example 1

Which of the figures below is a polygon?

The easiest way to identify the polygon is to identify which shapes are not polygons. Choices B and C each have at least one curved side. So they cannot be polygons. Choice D has all straight sides, but one of the vertices is not at the endpoints of the two adjacent sides, so it is not a polygon. Choice A is composed entirely of line segments that intersect at their endpoints. So, it is a polygon. The correct answer is A.

Example 2

Which of the figures below is not a polygon?

All four of the shapes are composed of line segments, so you cannot eliminate any choices based on that criteria alone. Notice that choices A, B, and D have points that all lie within the same plane. Choice C is a three-dimensional shape, so it does not lie within one plane. So it is not a polygon. The correct answer is C.

Convex and Concave Polygons

Now that you know how to identify polygons, you can begin to practice classifying them. The first type of classification to learn is whether a polygon is convex or concave. Think of the term concave as referring to a cave, or an interior space. A concave polygon has a section that “points inward” toward the middle of the shape. In any concave polygon, there are at least two vertices that can be connected without passing through the interior of the shape. The polygon below is concave and demonstrates this property.

A convex polygon does not share this property. Any time you connect the vertices of a convex polygon, the segments between nonadjacent vertices will travel through the interior of the shape. Lines segments that connect to vertices traveling only on the interior of the shape are called diagonals.

Example 3

Identify whether the shapes below are convex or concave.

To solve this problem, connect the vertices to see if the segments pass through the interior or exterior of the shape.

A. The segments go through the interior.

Therefore, the polygon is convex.

B. The segments go through the exterior.

Therefore, the polygon is concave.

C. One of the segments goes through the exterior.

Thus, the polygon is concave.

Classifying Polygons

The most common way to classify a polygon is by the number of sides. Regardless of whether the polygon is convex or concave, it can be named by the number of sides. The prefix in each name reveals the number of sides. The chart below shows names and samples of polygons.

Polygon Name	Number of Sides	Sample Drawings
Triangle	$3$
Quadrilateral	$4$
Pentagon	$5$
Hexagon	$6$
Heptagon	$7$
Octagon	$8$
Nonagon	$9$
Decagon	$10$
Undecagon or hendecagon (there is some debate!)	$11$
Dodecagon	$12$
$n-$ gon	$n$ (where $n>12$ )

Practice using these polygon names with the appropriate prefixes. The more you practice, the more you will remember.

Example 4

Name the three polygons below by their number of sides.

A. This shape has seven sides, so it is a heptagon.

B. This shape has five sides, so it is a pentagon.

C. This shape has ten sides, so it is a decagon.

Using the Distance Formula on Polygons

You can use the distance formula to find the lengths of sides of polygons if they are on a coordinate grid. Remember to carefully assign the values to the variables to ensure accuracy. Recall from algebra that you can find the distance between points $(x_1,y_1)$ and $(x_2,y_2)$ using the following formula.

$\mbox{Distance} = \sqrt{{(x_2 - x_1)}^2 + {(y_2 - y_1)}^2}$

Example 5

A quadrilateral has been drawn on the coordinate grid below.

What is the length of segment $BC$ ?

Use the distance formula to solve this problem. The endpoints of $\overline{BC}$ are $(-3,9)$ and $(4,1)$ . Substitute $-3$ for $x_1$ , $9$ for $y_1$ , $4$ for $x_2$ , and $1$ for $y_2$ . Then we have:

$D & = \sqrt{{(x_2 - x_1)}^2 + {(y_2 - y_1)}^2}\ D & = \sqrt{{(4 - (-3))}^2 + {(1 - 9)}^2}\ D & = \sqrt{(7)^2 + (-8)^2}\ D & = \sqrt{49 + 64}\ D & = \sqrt{113}$

So the distance between points $B$ and $C$ is $\sqrt{113}$ , or about $10.63\;\mathrm{units}$ .

Lesson Summary

In this lesson, we explored polygons. Specifically, we have learned:

How to define polygons.
How to understand the difference between convex and concave polygons.
How to classify polygons by number of sides.
How to use the distance formula to find side lengths on a coordinate grid.

Polygons are important geometric shapes, and there are many different types of questions that involve them. Polygons are important aspects of architecture and design and appear constantly in nature. Notice the polygons you see every day when you look at buildings, chopped vegetables, and even bookshelves. Make sure you practice the classifications of different polygons so that you can name them easily.

Review Questions

For exercises 1-5, name each polygon in as much detail as possible.

Explain why the following figures are NOT polygons:
How many diagonals can you draw from one vertex of a pentagon? Draw a sketch of your answer.
How many diagonals can you draw from one vertex of an octagon? Draw a sketch of your answer.
How many diagonals can you draw from one vertex of a dodecagon?
Use your answers to 7, 8, and 9 and try more examples if necessary to answer the question: How many diagonals can you draw from one vertex of an $n-$ gon?

Review Answers

This is a convex pentagon.
Concave octagon.
Concave $17-$ gon (note that the number of sides is equal to the number of vertices, so it may be easier to count the points [vertices] instead of the sides).
Concave decagon.
Convex quadrilateral.
A is not a polygon since the two sides do not meet at a vertex; B is not a polygon since one side is curved; C is not a polygon since it is not enclosed.
The answer is $2$ .
The answer is $5$ .
A dodecagon has twelve sides, so you can draw nine diagonals from one vertex.
Use this table to answer question 10.

Sides	Diagonals from One Vertex
$3$	$0$
$4$	$1$
$5$	$2$
$6$	$3$
$7$	$4$
$8$	$5$
$9$	$6$
$10$	$7$
$11$	$8$
$12$	$9$
$\ldots$	$\ldots$
$n$	$n-3$

To see the pattern, try adding a “process” column that takes you from the left column to the right side.

Sides	Process	Diagonals from One Vertex
$3$	$(3) - 3 = 0$	$0$
$4$	$(4) - 3 = 1$	$1$
$5$	$(5) - 3 = 2$	$2$
$6$	$(6) - 3 = 3$	$3$
$7$	$(7) - 3 = 4$	$4$
$8$	$(8) - 3 = 5$	$5$
$\ldots$		$\ldots$
$n$	$(n) - 3 =$	$n - 3$

Notice that we subtract
$3$
from each number on the left to arrive at the number in the right column. So, if the number in the left column is
$n$
(standing for some unknown number), then the number in the right column is
$n - 3$
.

Problem Solving in Geometry

Learning Objectives

Read and understand given problem situations.
Use multiple representations to restate problem situations.
Identify problem-solving plans.
Solve real-world problems using planning strategies.

Introduction

One of the most important things we hope you will learn in school is how to solve problems. In real life, problem solving is not usually as clear as it is in school. Often, performing a calculation or measurement can be a simple task. Knowing what to measure or solve for can be the greatest challenge in solving problems. This lesson helps you develop the skills needed to become a good problem solver.

Understanding Problem Situations

The first step whenever you approach a complicated problem is to simplify the problem. That means identifying the necessary information, and finding the desired value. Begin by asking yourself the simple question: What is this problem asking for?

If the problem had to ask you only one question, what would it be? This helps you identify how you should respond in the end.

Next, you have to find the information you need to solve the problem. Ask yourself another question: What do I need to know to find the answer?

This question will help you sift through information that may be helpful with this problem.

Use these basic questions to simplify the following problem. Don’t try to solve it yet, just begin this process with questioning.

Example 1

Ehab drew a rectangle $PQRS$ on the chalkboard. $PQ$ was $8 \;\mathrm{cm}$ and $QR$ was $6 \;\mathrm{cm}$ . If Ehab draws in the diagonal $\overline{QS}$ , what will be its length?

Begin to understand this problem by asking yourself two questions:

1. What is the problem asking for?

The question asks for the length of diagonal $\overline{QS}$ .

2. What do I need to know to find the answer?

You need to know three things:

The angles of a rectangle are all equal to $90^\circ$ .
The lengths of the sides of the rectangle are $8 \;\mathrm{cm}$ and $6 \;\mathrm{cm}$ .
The Pythagorean Theorem can be used to find the third side of a right triangle.

Answering these questions is the first step to success with this problem.

Drawing Representations

Up to this point, the analysis of the sample problem has dealt with words alone. It is important to distill the basic information from the problem, but there are different ways to proceed from here. Often, visual representations can be very helpful in understanding problems. Make a simple drawing that represents what is being discussed. For example, a tray with six cookies could be represented by the diagram below.

The drawing takes only seconds to create, but it could help you visualize important information. Remember that there are many different ways to display information. Look at the way a line segment six inches long is displayed below.

When you approach a problem, think about how you can represent the information in the most useful way. Continue your work on the sample problem by making drawings.

Let’s return to that example.

Example 1 (Repeated)

Ehab drew a rectangle $PQRS$ on the chalkboard. $PQ$ was $8 \;\mathrm{cm}$ and $QR$ was $6 \;\mathrm{cm}$ . If Ehab draws in the diagonal $\overline{QS}$ , what will be its length?

Think about the different ways in which you could draw the information in this problem. The simplest idea is to draw a labeled rectangle. Be sure to label your drawing with information from the problem. This includes the names of the vertices as well as the side lengths.

As in most situations that you will encounter, there is more than one correct way to draw this shape. Two more possibilities follow.

The first example above shows the internal structure of the rectangle, as it is divided into square centimeters. The second example shows the rectangle situated on a coordinate grid. Notice that we rotated the figure by $90^\circ$ in the second picture. This is fine as long as it was drawn maintaining side lengths. One implication of putting the figure on the coordinate grid is that one square unit on the grid is equivalent to one square centimeter.

Identifying Your Strategy

At this point, you have simplified the problem by asking yourself questions about it, and created different representations of the important information. The time has come to establish a formal plan of attack. This is a crucial step in the problem-solving process, as it lays the groundwork for your solution.

To organize your thoughts, think of your geometric knowledge as a toolbox. Each time you learn a new strategy, technique, or concept, add it to your toolbox. Then, when you need to solve a problem, you can select the appropriate tool to use.

For now, take a quick look at the representations drawn for the example problem to identify what tools you might need. You can use this section to clearly identify your strategy.

Example

Ehab drew a rectangle $PQRS$ on the chalkboard. $PQ$ was $8 \;\mathrm{cm}$ and $QR$ was $6 \;\mathrm{cm}$ . If Ehab draws in the diagonal $\overline{QS}$ , what will be its length?

In the first representation, there is simply a rectangle with a diagonal. Though there is a way to solve this problem using this diagram, it will not be covered until later in this book. For now, you do not have the tools to solve it.

The second diagram shows the building blocks that comprise the rectangle. The diagonal cuts through the blocks but presents the same challenges as the first diagram. You do not yet have the tools to solve the problem using this diagram either.

The third diagram shows a coordinate grid with the rectangle drawn in. The diagonal has two endpoints with specific coordinate pairs. In this chapter, you learned the distance formula to find lengths on a coordinate grid. This is the tool you need to solve the problem.

Your strategy for this problem is to identify the two endpoints of $QS$ on the grid as $(x_1,y_1)$ and $(x_2,y_2)$ . Use the distance formula to find the length. The result will be the solution to the problem.

Making Calculations

The last step in any problem-solving situation is employing your strategy to find the answer. Be sure that you use the correct values as identified in the relevant information. When you perform calculations, use a pencil and paper to keep track of your work. Many careless mistakes result from mental calculations. Keep track of each step along the way.

Finally, when you have found the answer, there are two more questions to ask yourself:

1. Did I provide the information the problem requested?

Go back to the first stages of the problem. Verify that you answered all parts of the question.

2. Does my answer make sense?

Your answer should make sense in the context of the problem. If your number is abnormally large or small in value, check your work.

Example

Ehab drew a rectangle $PQRS$ on the chalkboard. $PQ$ was $8 \;\mathrm{cm}$ and $WR$ was $6 \;\mathrm{cm}$ . If Ehab draws in the diagonal $\overline{QS}$ , what will be its length?

At this point, we have distilled the problem, created multiple representations of the scenario, and identified the desired strategy. It is time to solve the problem.

The diagram below shows the rectangle on the coordinate grid.

To find the length of $\overline{QS}$ , you must identify its endpoints on the grid. They are $(1,1)$ and $(9,7)$ . Use the distance formula and substitute $1$ for $x_1$ , $1$ for $y_1$ , $9$ for $x_2$ , and $7$ for $y_2$ .

$\text{distance} & = \sqrt{{(x_2 - x_1)}^2 + {(y_2 - y_1)}^2}\ \text{distance} & = \sqrt{{(9 - 1)}^2 + {(7 - 1)}^2}\ \text{distance} & = \sqrt{(8)^2 + (6)^2}\ \text{distance} & = \sqrt{64 + 36}\ \text{distance} & = \sqrt{100}\ \text{distance} & = 10$

$QS$ is $10 \;\mathrm{cm}$ .

Finally, make sure to ask yourself two more questions to verify your answer.

1. Did I provide the information the problem requested?

The problem asked you to identify the length of $\overline{QS}$ . That is the information provided with our solution.

2. Does my answer make sense?

The value of $10 \;\mathrm{cm}$ is slightly larger than $6 \;\mathrm{cm}$ or $8 \;\mathrm{cm}$ , but that is to be expected in this scenario. It is certainly within reason. A response of $80 \;\mathrm{cm}$ or $0.08 \;\mathrm{cm}$ would have been unreasonable.

Your work on this problem is now complete. The final answer is $10 \;\mathrm{cm}$ .

Lesson Summary

In this lesson, we explored problem-solving strategies. Specifically, we have learned:

How to read and understand given problem situations.
How to use multiple representations to restate problem situations.
How to identify problem-solving plans.
How to solve real-world problems using planning strategies.

These skills are important for any type of problem, whether or not it is about geometry. Practice breaking down different problems in other parts of your life using these techniques. Forming plans and using strategies will help you in a number of different ways.

Points to Consider

This chapter focused on the basic postulates of geometry and the most common vocabulary and notations used throughout geometry. The following chapters focus on the skills of logic, reasoning, and proof. Review the material in this chapter whenever necessary to maintain your understanding of the basic geometric principles. They will be necessary as you continue in your studies.

Review Questions

Suppose one line is drawn in a plane. How many regions of the plane are created?
Suppose two lines intersect in a plane. How many regions is the plane divided into? Draw a diagram of your answer.
Now suppose three coplanar lines intersect at the same point in a plane. How many regions is the plane divided into? Draw a diagram of your answer.
Make a table for the case of $4, 5, 6,$ and $7$ coplanar lines intersecting at one point.
Generalize your answer for number 4. If $n$ coplanar lines intersect at one point, the plane is divided into __________ regions.
Bindi lives twelve miles south of Cindy. Mari lives five miles east of Bindi. What is the distance between Cindy's house and Mari’s house?
1. Model this problem by drawing it on a coordinate grid. Let Bindi’s house bet at the origin, $(0,0)$ . Use the labels $B$ for Bindi’s house, $M$ for Mari’s house, and $C$ for Cindy’s house.
2. What are the coordinates of Cindy's and Mari’s house?
3. Use the distance formula to find the distance between
Suppose a camper is standing $100\;\mathrm{meters}$ north of a river that runs east-west in a perfectly straight line (we have to make some assumptions for geometric modeling!). Her tent is $25\;\mathrm{meters}$ north of the river, but $300\;\mathrm{meters}$ downstream. See the diagram below).

The camper sees that her tent has caught fire! Luckily she is carrying a bucket so she can get water from the river to douse the flames. The camper will run from her current position to the river, pick up a bucket of water, and then run to her tent to douse the flames (see the blue line in the diagram). But how far along the river should she run (distance $x$ in the diagram) to pick up the bucket of water if she wants to minimize the total distance she runs? Solve this by any means you see fit—use a scale model, the distance formula, or some other geometric method.

Does it make sense for the camper in problem 7 to want to minimize the total distance she runs? Make an argument for or against this assumption. (Note that in real-life problem solving finding the “best” answer is not always simple!).

Review Answers

$2$
$4$
$6$
See the table below

Number of Coplanar Lines Intersecting at One Point	Number of Regions Plane is Divided Into
$1$	$2$
$2$	$4$
$3$	$6$
$4$	$8$
$5$	$10$
$6$	$12$
$7$	$14$

Every number in the right-hand column is two times the number in the left-hand column, so the general statement is: “If $n$ coplanar lines intersect at one point, the plane is divided into $2n$ regions.”
2. Cindy’s House: $(0,12)$ ; Mari’s house: $(5,0)$
3. $13\;\mathrm{miles}$
One way to solve this is to use a scale model and a ruler. Let $1 \;\mathrm{cm} = 100 \;\mathrm{m}$ . Then you can draw a picture and measure the distance the camper has to run for various locations of the point where she gets water. Be careful using the scale!

Now make a table for all measurements to find the best, shortest total distance.

$x\;\mathrm{(meters)}$	Distance to Water $(m)$	Distance from Water to Tent $(m)$	Total Distance $(m)$
$0$	$100$	$301$	$401$
$25$	$103$	$276$	$379$
$50$	$112$	$251$	$363$
$100$	$141$	$202$	$343$
$125$	$160$	$177$	$337$
$150$	$180$	$152$	$332$
$175$	$202$	$127$	$329$
$200$	$224$	$103$	$327$
$225$	$246$	$79$	$325$
$250$	$269$	$56$	$325$
$275$	$293$	$35$	$328$
$300$	$316$	$25$	$341$

It looks like the best place to stop is between $225$ and $250\;\mathrm{meters}$ . Based on other methods (which you will learn in calculus and some you will learn later in geometry), we can prove that the best distance is when she runs $240\;\mathrm{meters}$ downstream to pick up the bucket of water.

Answers will vary. One argument for why it is not best to minimize total distance is that she may run slower with the full bucket of water, so she should take the distance she must run with a full bucket into account.