Geometry Workbook

Welcome to the Ready, Set, Go! Geometry Workbook!

About This Book

This book will help high school math students at all learning levels understand basic geometry. Students will develop the skills, confidence, and knowledge they need to succeed on high school math exams with emphasis on passing high school graduation exams.

More than 20 easy-to-follow lessons break down the material into the basics. In-depth, step-by-step examples and solutions reinforce student learning, while the Math Flash feature provides useful tips and strategies, including advice on common mistakes to avoid.

Students can take drills and quizzes to test themselves on the subject matter, then review any areas in which they need improvement or additional reinforcement. The book concludes with a final exam, designed to comprehensively test what students have learned.

The Ready, Set, Go! Geometry Workbook will help students master the basics of mathematics—and help them face their next math test—with confidence!

Icons Explained

Icons make navigating through the book easier. The icons, explained below, highlight tips and strategies, where to review a topic, and the drills found at the end of each lesson.

Look for the "Math Flash feature for helpful tips and strategies, including advice on how to avoid common mistakes.

When you see the "Let’s Review" icon, you know just where to look for more help with the topic on which you are currently working.

The "Test Yourself!" icon, found at the end of every lesson, signals a short drill that reviews the skills you have studied in that lesson.

To the Student

This workbook will help you master the fundamentals of Algebra & Functions. It offers you the support you need to boost your skills and helps you succeed in school and beyond!

It takes the guesswork out of math by explaining what you most need to know in a step-by-step format. When you apply what you learn from this workbook, you can

do better in class;

raise your grades, and

score higher on your high school math exams.

Each compact lesson in this book introduces a math concept and explains the method behind it in plain language. This is followed with lots of examples with fully worked-out solutions that take you through the key points of each problem.

The book gives you two tools to measure what you learn along the way:

e9780738667416_img_10003.gif Short drills that followeachlesson

e9780738667416_img_10003.gif Quizzes that test you onmultiplelessons

These tools are designed to comfortably build your test-taking confidence.

Meanwhile, the Math Flash feature throughout the book offers helpful tips and strategies—including advice on how to avoid common mistakes.

When you complete the lessons, take the final exam at the end of the workbook to see how far you’ve come. If you still need to strengthen your grasp on any concept, you can always go back to the related lesson and review at your own pace.

To the Parent

For many students, math can be a chahenge—but with the right tools and support, your child can master the basics of algebra. As educational publishers, our goal is to help all students develop the crucial math skills they’ll need in school and beyond.

This Ready, Set, Go! Workbook is intended for students who need to build their basic geometry skills. It was specifically created and designed to assist students who need a boost in understanding and learning the math concepts that are most tested along the path to graduation. Through a series of easy-to-follow lessons, students are introduced to the essential mathematical ideas and methods, and then take short quizzes to test what they are learning.

Each lesson is devoted to a key mathematical building block. The concepts and methods are fully explained, then reinforced with examples and detailed solutions. Your child will be able to test what he or she has learned along the way, and then take a cumulative exam found at the end of the book.

Whether used in school with teachers, for home study, or with a tutor, the Ready, Set, Go! Workbook is a great support tool. It can help improve your child’s math proficiency in a way that’s fun and educational!

To the Teacher

As you know, not all students learn the same, or at the same pace. And most students require additional instruction, guidance, and support in order to do well academically.

Using the Curriculum Focal Points of the National Council of Teachers of Mathematics, this workbook was created to help students increase their math abilities and succeed on high school exams with special emphasis on high school proficiency exams. The book’s easy-to-follow lessons offer a review of the basic material, supported by examples and detailed solutions that illustrate and reinforce what the students have learned.

To accommodate different pacing for students, we provide drills and quizzes throughout the book to enable students to mark their progress. This approach allows for the mastery of smaller chunks of material and provides a greater opportunity to build mathematical competence and confidence.

When we field-tested this series in the classroom, we made every effort to ensure that the book would accommodate the common need to build basic math skills as effectively and flexibly as possible. Therefore, this book can be used in conjunction with lesson plans, stand alone as a single teaching source, or be used in a group-learning environment. The practice quizzes and drills can be given in the classroom as part of the overall curriculum or used for independent study. A cumulative exam at the end of the workbook helps students (and their instructors) gauge their mastery of the subject matter.

We are confident that this workbook will help your students develop the necessary skills and build the confidence they need to succeed on high school math exams.

About Research & Education Association

Founded in 1959, Research & Education Association (REA) is dedicated to publishing the finest and most effective educational materials—including software, study guides, and test preps—for students in elementary school, middle school, high school, college, graduate school, and beyond.

Today, REA’s wide-ranging catalog is a leading resource for teachers, students, and professionals.

We invite you to visit us at www.rea.com to find out how REA is making the world smarter.

About the Author

Author Mel Friedman is a former classroom teacher and test-item writer for Educational Testing Service and ACT, Inc.

Acknowledgments

We would like to thank Larry Kling, Vice President, Editorial, for his editorial direction; Pam Weston, Vice President, Publishing, for setting the quality standards for production integrity and managing the publication to completion; Alice Leonard, Senior Editor, for project management and preflight editorial review; Diane Goldschmidt, Senior Editor, for post-production quality assurance; Ruth O’Toole, Production Editor, for proofreading; Rachel DiMatteo, Graphic Artist, for page design; Christine Saul, Senior Graphic Artist, for cover design; and Jeff LoBalbo, Senior Graphic Artist, for post-production file mapping

We also gratefully acknowledge Heather Brashear for copyediting, and Kathy Caratozzolo of Caragraphics for typesetting.

Thank you to author Bob Miller for his review of all test items.

Orsolina Cetta, Jamie Chaikin and the students of Piscataway High School, Piscataway, NJ, for reviewing and field-testing lessons from this book.

1

Basic Properties of Points, Rays, Lines, and Angles

In this lesson, we will look at the features and properties of points, lines, line segments, rays, and angles. Each lesson will use at least one of these elements, and many lessons will use all five of them. In real life, we see these elements everywhere. We find them, for example, as (a) the tip of a pen, (b) an arrow, (c) a piece of string, and (d) the path of a billiard ball in a game of pool.

Your Goal: When you have completed this lesson, you should be familiar with the properties of points, rays, lines, and angles. In addition, you will be able to correctly name each of them, using the correct letters.

All geometric shapes that you will meet in this workbook are considered to lie in a single plane. A plane is essentially a flat surface with length and width, but no depth. A plane extends indefinitely. A good way to picture a plane is to think of a magic carpet that extends in all directions but has no thickness.

The most basic unit in geometry is a point. Examples are pencil points, tiny seeds, and dots. A point has location, but no dimension; thus, it is infinitely small. Points are named by using a capital letter. Here are examples of point P, point Q, and point R.

Technically, these drawn points have some length and width (although very small). However, they are only visual representations of points.

A line is a row of points that can all be connected with a ruler. Lines are named by using any two of its points, written in capital letters, with the symbol ↔ over the letters that name or identify the two points. The letters used do not have to be in alphabetical order. Here are examples, with notation, of line AB, line DC, and line EH:

A line segment is a finite portion of a line. Finite means it has a beginning and an end. It has two endpoints that are named with capital letters, with the bar symbol over the two letters. As with lines, the letters of the points used do not have to be in alphabetical order. Here are examples, with notation, of line segments JK, NP, and XY:

Lines and line segments may face in any direction, but the notation over the associated letters is always horizontal. Both lines and line segments have length, but no width. Line segments have measurable length, whereas the length of a line cannot be measured.

A ray is basically a half line in that it consists of a starting point, called its end point, and goes forever in the direction of a line. It is named by using its end point first, followed by any other point that it contains. The symbol → is placed over the two capital letters that are used to denote the ray. Rays have length, but no width. Here are examples, with notation, of rays MQ, SR, and VT:

With lines or line segments, the letters used to name them can be reversed, so that e9780738667416_i0013.jpg means the same as e9780738667416_i0014.jpg With rays, the first letter used must be its end point. Rays may face in any direction, but the notation over the letters used is always horizontal.

An angle consists of two rays that share a common end point. The only restriction is that one of the rays must not overlap the other. The common end point is called the vertex. (The plural of vertex is vertices.) The two rays that form the angle are called sides. Using a protractor, an angle can be measured by the opening between its rays. The unit of measurement is commonly called a degree. An angle is named in one of three ways:

by its vertex;

by a point on one ray, followed by its vertex and then a point on the other ray; and

by a number.

The symbol Z is placed before the letter(s) used to name an angle. Here are examples, with notation, of angle F, angle SUV, and angle 1:

The symbol ∠ is used even if the angle appears larger. In addition, an angle may actually be a line, since the two rays do not overlap. Numbers used to name angles are generally low integers. So if a diagram had three angles next to each other, they would usually be named ∠1, ∠2, and ∠3. Here is the associated diagram.

1 Example: How can the following line be named in three different ways?

Solution: Using any two letters in any order, we could write e9780738667416_i0019.jpg or e9780738667416_i0020.jpg

2 Example: In how many different ways can the following line segment be named?

Solution: There are only two ways, namely, e9780738667416_i0022.jpg or e9780738667416_i0023.jpg

3 Example: In how many different ways can the following angle be named?

Solution: There are three ways, namely, ∠P, ∠OPN, and ∠NPO.

4 Example: In the following diagram, in how many other ways can ∠1 be named?

Solution: There are two ways, namely, ∠CAD and ∠DAC.

In Example 4, you cannot use ∠A to name ∠1. In fact, the symbol ∠A cannot be used for any of the angles shown, since there would be confusion as to which angle you are talking about. There are actually three distinct angles with a vertex at A. Also, ∠1 and ∠2 are called adjacent angles, since they share the same vertex and a common ray.

5 Example: In the following diagram, which angle is adjacent to ∠FGK?

Solution: ∠KGJ is adjacent to ∠FGK, since they share the vertex G and the common ray GK.

6 Example: In the following diagram, how do you name the rays that form ∠3?

Solution: The two rays forming ∠3 are e9780738667416_i0029.jpg and e9780738667416_i0030.jpg

7 Example: In the following diagram, which angles may be named using only the vertex?

Solution: Only ∠X and ∠Z can be named using only the vertex. At each of points W and Y there are actually three angles.

8 Example: Returning to Example 7, how do you name the three angles at Y?

Solution: The largest angle at Y is ∠ZYX. The two smaller angles are ∠ZYW and ∠XYW.

9 Example: How do you name the following ray in three different ways?

Solution: The three ways are e9780738667416_i0033.jpg and e9780738667416_i0034.jpg

10 Example: How do you name the following angle in six different ways?

Solution: The six ways are ∠4, ∠Q, ∠ZQT, ∠ZQV, ∠TQZ, and ∠VQZ.

How many rays are needed to define an angle?

Answer: ______________________________

How many points does a line have?

Answer: ______________________________

How do you name the following angle in three ways?

Answers:______________________________

How do you name the angle that is adjacent to ∠YSN?

Answer:______________________________

In the following diagram, how do you name the three different line segments?

Answers:______________________________

Which one of the following would most closely resemble a ray?

Paper clip

One-way street sign

Stop sign

Long wire

Which one of the following correctly describes a line segment?

Length and width

Width, but no length

Neither length nor width

Length, but no width

Look at the following figure.

Which angle(s) can be correctly described using only the vertex letter?

∠G, ∠H, and ∠K

∠G, ∠H, ∠I, ∠J, and ∠K

∠I, and ∠J

None of the angles

How many letters are needed to identify a point?

Answer: ______________________________

How do you name the following angle in five different ways?

Answers:______________________________

An angle is defined as the formation of two rays. But, within a geometric figure such as the one that appears in question #8, angles are shown as the formation of two line segments. The entire ∠K would be seen from this diagram.

2

Measuring Line Segments and Angles

In this lesson, we will explore how to measure line segments and angles using their appropriate units. We cannot measure points, lines, or rays. Remember that points have no dimension, and both rays and lines extend indefinitely. A line segment has a specific length, and an angle has a specific opening.

Your Goal: When you have completed this lesson, you should be familiar with the ways in which line segments and angles are measured.

Given a line segment, e9780738667416_i0046.jpg we can simply calculate, using a ruler, the number of inches between points A and B. This is called the distance of e9780738667416_i0047.jpg and is denoted as AB or BA.

In most calculation problems concerning line segments, line segments will have numbers artificially assigned to them. This means that if you see a line segment that has the number 4 associated with it, you would say that its length is 4 units, where a unit is some specific (but arbitrary) length. This assigned number is called the measure of the line segment.

To clarify this concept, you may see e9780738667416_i0048.jpg in one problem, but then see e9780738667416_i0049.jpg in another problem. Don’t be disturbed that e9780738667416_i0050.jpg has a higher number than e9780738667416_i0051.jpg The length of the unit used in each case is different. However,