Geometry - Allen Ma - E-Book

Geometry E-Book

Allen Ma

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Beschreibung

Practice makes perfect! Get perfect with a thousand and one practice problems! 1,001 Geometry Practice Problems For Dummies gives you 1,001 opportunities to practice solving problems that deal with core Geometry topics, such as points, lines, angles, and planes, as well as area and volume of shapes. You'll also find practice problems on more advanced topics, such as proofs, theorems, and postulates. The companion website gives you free online access to 500 practice problems and solutions. You can track your progress and ID where you should focus your study time. The online component works in conjunction with the book to help you polish your skills and build confidence. As the perfect companion to Geometry For Dummies or a stand-alone practice tool for students, this book & website will help you put your Geometry skills into practice, encouraging deeper understanding and retention. The companion website includes: * Hundreds of practice problems * Customizable practice sets for self-directed study * Problems ranked as easy, medium, and hard * Free one-year access to the online questions bank With 1,001 Geometry Practice Problems For Dummies, you'll get the practice you need to master Geometry and gain confidence in the classroom.

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1,001 Geometry Practice Problems For Dummies®

Published by: John Wiley & Sons, Inc., 111 River Street, Hoboken, NJ 07030-5774, www.wiley.com

Copyright © 2015 by John Wiley & Sons, Inc., Hoboken, New Jersey

Media and software compilation copyright © 2015 by John Wiley & Sons, Inc. All rights reserved.

Published simultaneously in Canada

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Library of Congress Control Number: 20149456253

ISBN 978-1-118-85326-9 (pbk); ISBN 978-1-118-85305-4 (ebk); ISBN 978-1-118-85302-3 (ebk)

1,001 Geometry Practice Problems For Dummies®

Visit www.dummies.com/cheatsheet/1001geometry to view this book's cheat sheet.

Table of Contents

Cover

Introduction

What You’ll Find

Beyond the Book

Where to Go for Additional Help

Part I: The Questions

Chapter 1: Diving into Geometry

The Problems You’ll Work On

What to Watch Out For

Chapter 2: Constructions

The Problems You’ll Work On

What to Watch Out For

Chapter 3: Geometric Proofs with Triangles

The Problems You’ll Work On

What to Watch Out For

Chapter 4: Classifying Triangles

The Problems You’ll Work On

What to Watch Out For

Chapter 5: Investigating the Centers of a Triangle

The Problems You’ll Work On

What to Watch Out For

Chapter 6: Similar Triangles

The Problems You’ll Work On

What to Watch Out For

Chapter 7: The Right Triangle

The Problems You’ll Work On

What to Watch Out For

Chapter 8: Triangle Inequalities

The Problems You’ll Work On

What to Watch Out For

Chapter 9: Polygons

The Problems You’ll Work On

What to Watch Out For

Chapter 10: Properties of Parallel Lines

The Problems You’ll Work On

What to Watch Out For

Chapter 11: Properties of Quadrilaterals

The Problems You’ll Work On

What to Watch Out For

Chapter 12: Coordinate Geometry

The Problems You’ll Work On

What to Watch Out For

Chapter 13: Transformational Geometry

The Problems You’ll Work On

What to Watch Out For

Chapter 14: Exploring Circles

The Problems You’ll Work On

What to Watch Out For

Chapter 15: Circle Theorems

The Problems You’ll Work On

What to Watch Out For

Chapter 16: Three-Dimensional Geometry

The Problems You’ll Work On

What to Watch Out For

Chapter 17: Locus Problems

The Problems You’ll Work On

What to Watch Out For

Part II: The Answers

Chapter 18: Answers and Explanations

About the Authors

Cheat Sheet

Connect with Dummies

End User License Agreement

Guide

Cover

Table of Contents

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Introduction

This book is intended for anyone who needs to brush up on geometry. You may use this book as a supplement to material you’re learning in an undergraduate geometry course. The book provides a basic level of geometric knowledge. As soon as you understand these concepts, you can move on to more complex geometry problems.

What You’ll Find

The 1,001 geometry problems are grouped into 17 chapters. You’ll find calculation questions, construction questions, and geometric proofs, all with detailed answer explanations. If you miss a question, take a close look at the answer explanation. Understanding where you went wrong will help you learn the concepts.

Beyond the Book

This book provides a lot of geometry practice. If you’d also like to track your progress online, you’re in luck! Your book purchase comes with a free one-year subscription to all 1,001 practice questions online. You can access the content with your computer, tablet, or smartphone whenever you want. Create your own question sets and view personalized reports that show what you need to study most.

What you’ll find online

The online practice that comes free with the book contains the same 1,001 questions and answers that are available in the text. You can customize your online practice to focus on specific areas, or you can select a broad variety of topics to work on — it’s up to you. The online program keeps track of the questions you get right and wrong so you can easily monitor your progress.

This product also comes with an online Cheat Sheet that helps you increase your geometry knowledge. Check out the free Cheat Sheet at (www.dummies.com/cheatsheet/1001geometry) (No PIN required. You can access this info before you even register.)

How to register

To gain access to additional tests and practice online, all you have to do is register. Just follow these simple steps:

Find your PIN access code:

Print-book users:

If you purchased a print copy of this book, turn to the inside front cover of the book to find your access code.

E-book users:

If you purchased this book as an e-book, you can get your access code by registering your e-book at

www.dummies.com/go/getaccess

. Go to this website, find your book and click it, and answer the security questions to verify your purchase. You’ll receive an email with your access code.

Go to

Dummies.com

and click

Activate Now.

Find your product (

1,001 Geometry Practice Problems For Dummies (+ Free Online Practice)

) and then follow the on-screen prompts to activate your PIN.

Now you’re ready to go! You can come back to the program as often as you want — simply log on with the username and password you created during your initial login. No need to enter the access code a second time.

For Technical Support, please visit http://wiley.custhelp.com or call Wiley at 1-800-762-2974 (U.S.), +1-317-572-3994 (international).

Where to Go for Additional Help

This book covers a great deal of geometry material. Because there are so many topics, you may struggle in some areas. If you get stuck, consider getting some additional help.

In addition to getting help from your friends, teachers, or coworkers, you can find a variety of great materials online. If you have Internet access, a simple search often turns up a treasure trove of information. You can also head to www.dummies.com to see the many articles and books that can help you in your studies.

1,001 Geometry Questions For Dummies gives you just that — 1,001 practice questions and answers to improve your understanding and application of geometry concepts. If you need more in-depth study and direction for your geometry courses, you may want to try out the following For Dummies products:

Geometry For Dummies, by Mark Ryan:

This book provides an introduction into the most important geometry concepts. You’ll learn all the principles and formulas you need to analyze two- and three-dimensional shapes. You’ll also learn the skills and strategies needed to write a geometric proof.

Geometry Workbook For Dummies, by Mark Ryan:

This workbook guides you through geometric proofs using a step-by-step process. It also provides tips, shortcuts, and mnemonic devices to help you commit some important geometry concepts to memory.

Part I

The Questions

Visit www.dummies.com for free access to great For Dummies content online.

In this part …

The best way to become proficient in geometry is through a lot of practice. Fortunately, you now have 1,001 practice opportunities right in front of you. These questions cover a variety of geometric concepts and range in difficulty from easy to hard. Master these problems, and you’ll be well on your way to a solid foundation in geometry.

Here are the types of problems that you can expect to see:

Geometric definitions (

Chapter 1

)

Constructions (

Chapter 2

)

Geometric proofs with triangles (

Chapter 3

)

Classifying triangles (

Chapter 4

)

Centers of a triangle (

Chapter 5

)

Similar triangles (

Chapter 6

)

The Pythagorean theorem and trigonometric ratios (

Chapter 7

)

Triangle inequality theorems (

Chapter 8

)

Polygons (

Chapter 9

)

Parallel lines cut by a transversal (

Chapter 10

)

Quadrilaterals (

Chapter 11

)

Coordinate geometry (

Chapter 12)

Transformations (

Chapter 13

)

Circles (

Chapters 14

and

15

)

Surface area and volume of solid figures (

Chapter 16

)

Loci (

Chapter 17

)

Chapter 1

Diving into Geometry

Geometry requires you to know and understand many definitions, properties, and postulates. If you don’t understand these important concepts, geometry will seem extremely difficult. This chapter provides practice with the most important geometric properties, postulates, and definitions you need in order to get started.

The Problems You’ll Work On

In this chapter, you see a variety of geometry problems. Here’s what they cover:

Understanding midpoint, segment bisectors, angle bisectors, median, and altitude

Working with the properties of perpendicular lines, right angles, vertical angles, adjacent angles, and angles that form linear pairs

Noting the differences between complementary and supplementary angles

Using the addition and subtraction postulates

Understanding the reflexive, transitive, and substitution properties

What to Watch Out For

The following tips may help you avoid common mistakes:

Be on the lookout for when something is being done to a segment or an angle. Bisecting a segment creates two congruent segments, whereas bisecting an angle creates two congruent angles.

The transitive property and the substitution property look extremely similar in proofs, making them very confusing. Check whether you’re just switching the congruent segments/angles or whether you’re getting a third set of congruent segments/angles after already being given two pairs of congruent segments/angles.

Make sure you understand what the question is asking you to solve for. Sometimes a question asks only for a particular variable, so as soon as you find the variable, you’re done. However, sometimes a question asks for the measure of the segment or angle; after you find the value of the variable, you have to plug it in to find the measure of the segment or angle.

Understanding Basic Geometric Definitions

1–3 Fill in the blank to create an appropriate conclusion to the given statement.

1. If M is the midpoint of , then .

2. If bisects at E, then .

3. If , then _____ is a right angle.

4–9 In the following figure, bisects and . Determine whether each statement is true or false.

4. is a right angle.

5. .

6. and form a linear pair.

7. .

8. is an obtuse angle.

9. If Point S is the midpoint of , then it’s always true that .

10–14 Use the following figure and the given information to draw a valid conclusion.

10. is the median of .

11. is the altitude of .

12. bisects .

13. F is the midpoint of .

14. F is the midpoint of . What type of angle does have to be in order for to be called a perpendicular bisector?

Applying Algebra to Basic Geometric Definitions

15–18 Use the figure and the given information to answer each question.

15. E is the midpoint of . If and , find the value of x.

16. bisects . If is represented by and is represented by , find .

17. If and is represented by , find the value of x.

18. bisects . If and , find the length of .

Recognizing Geometric Terms

19–26 Write the geometric term that fits the definition.

19. Two adjacent angles whose sum is a straight angle: _______________

20. Two lines that intersect to form right angles: _______________

21. An angle whose measure is between 0° and 90°: _______________

22. A type of triangle that has two sides congruent and the angles opposite them also congruent: _______________

23. Divides a line segment or an angle into two congruent parts: _______________

24. An angle greater than 90° but less than 180°: _______________

25. A line segment connecting the vertex of a triangle to the midpoint of the opposite side: _______________

26. The height of a triangle: _______________

Properties and Postulates

27–34 Refer to segment to fill in the blank.

27.

28.

29.

30.

31. The _______________ would be the reason used to prove that .

32. If , then .

33. If , then .

34. Assuming the figure is not drawn to scale, if and , then you can prove that . The _______________ postulate can be used to draw this conclusion.

35–40 In the given diagram, . Use the basic geometric postulates to answer each question.

35. Which property or postulate is used to show that ?

36.

37.

38. What information must be given in order for the following to be true?

39. If bisects , you can conclude that .

40. If bisects , you can conclude that .

Adjacent Angles, Vertical Angles, and Angles That Form Linear Pairs

41–47 In the following figure,

Chapter 2

Constructions

One of the most visual topics in geometry is constructions. In this chapter, you get to demonstrate some of the most important geometric properties and definitions using a pencil, straight edge, and compass.

The Problems You’ll Work On

In this chapter, you see a variety of construction problems:

Constructing congruent segments and angles

Drawing segment, angle, and perpendicular bisectors

Creating constructions involving parallel and perpendicular lines

Constructing

and

triangles

What to Watch Out For

The following tips may help you avoid common mistakes:

If you’re drawing two arcs for a construction, make sure you keep the width of the compass (or radii of the circles) consistent.

Make your arcs large enough so that they intersect.

Sometimes you need to do more than one construction to create what the problem is asking for. This idea is extremely helpful when you need to construct special triangles.

Creating Congruent Constructions

61–65 Use your knowledge of constructions (as well as a compass and straight edge) to create congruent segments, angles, or triangles.

61. Construct , a line segment congruent to .

62. Construct , an angle congruent to .

63. Construct , a triangle congruent to .

64. Is the following construction an angle bisector or a copy of an angle?

65. Construct , a triangle congruent to .

Constructions Involving Angles and Segments

66–70 Apply your knowledge of constructions to angles and segments.

66. Construct segment , whose measure is twice the measure of .

67. Given , construct , the bisector of .

68. Construct the angle bisector of .

69. What type of construction is represented by the following figure?

70. True or False? The construction in the following diagram proves that .

Parallel and Perpendicular Lines

71–77 Apply your knowledge of constructions to problems involving parallel and perpendicular lines.

71. Place Point E anywhere on . Construct perpendicular to through Point E.

72. Use the following diagram to construct a line perpendicular to through Point C.

73. Construct the perpendicular bisector of .

74. Which construction is represented in the following figure?

75. Construct a line parallel to that passes through Point C.

76. True or False? The construction in the following diagram proves that .

77. True or False? The following diagram is the correct illustration of the construction of a line parallel to

Chapter 3

Geometric Proofs with Triangles

In geometry, you’re frequently asked to prove something. In this chapter, you’re given specific information and asked to prove specific information about triangles. You do this by using various geometric properties, postulates, and definitions to generate new statements that will lead you toward the information you’re looking to prove true.

The Problems You’ll Work On

In this chapter, you see a variety of problems involving geometric proofs:

Using SAS, SSS, ASA, and AAS to prove triangles congruent

Showing that corresponding parts of congruent triangles are congruent

Formulating a geometric proof with overlapping triangles

Using your knowledge of quadrilaterals to complete a geometric proof

Completing indirect proofs

What to Watch Out For

Remember the following tips as you work through this chapter:

The statement that needs to be proven has to be the last statement of the proof. It can’t be used as a given statement.

You must use all given information to formulate the proof. Each given should be used separately to draw its own conclusion.

If you’ve used all your given information and still require more to prove the triangles congruent, look for the reflexive property or a pair of vertical angles.

After you find angles or segments congruent, mark them in your diagram. The markings make it easier for you to see what other information you need to complete the proof.

To prove parts of a triangle congruent, you’ll first need to prove that the triangles are congruent to each other using the proper triangle congruence theorems.

Completing Geometric Proofs Using Triangle Congruence Theorems

103-107 Use the following figure to answer each question.

Given: and bisect each other at B.

Prove:

Statements

Reasons

1. and bisect each other at B.

1. Given

2. and

2.

3.

3. Intersecting lines form vertical angles.

4.

4.

5.

5.

6.

6.

103. What is the reason for Statement 2?

104. What is the statement for Reason 3?

105. What is the reason for Statement 4?

106. What is the reason for Statement 5?

107. What is the reason for Statement 6?

108–111 Use the following figure to answer each question.

Given: , , and

Prove:

Statements

Reasons

1. , ,

1. Given

2.

2.

3.

3.

4.

4.

5.

5.

108. What is the reason for Statement 2?

109. What is the reason for Statement 3?

110. What is the reason for Statement 4?

111. What is the reason for Statement 5?

112–116 Use the following figure to answer each question.

Given: and

Prove:

Statements

Reasons

1. and

1. Given

2.

2. When two parallel lines are cut by a transversal, alternate interior angles are formed.

3.

3.

4.

4. Reflexive property

5.

5.

6.

6.

112. What is the statement for Reason 2?

113. What is the reason for Statement 3?

114. What is the statement for Reason 4?

115. What is the reason for Statement 5?

116. What is the reason for Statement 6?

117 Complete the following proof.

117.

Given: , and M is the midpoint of .

Prove:

Overlapping Triangle Proofs

118–120 Use the following figure to answer the question regarding overlapping triangles.

Given: and

Prove:

Statements

Reasons

1. and

1. Given

2.

2.

3.

3.

4.

4.

118. What is the reason for Statement 2?

119. What is the reason for Statement 3?

120. What is the reason for Statement 4?

121–125 Use the following figure to answer each question regarding overlapping triangles.

Given: , , and

Prove:

Statements

Reasons

1. , , and

1. Given.

2.

2. Perpendicular lines form right angles.

3.

3.

4.

4.

5.

5.

6.

6.

121. What is the statement for Reason 2?

122. What is the reason for Statement 3?

123. What is the reason for Statement 4?

124.

Chapter 4

Classifying Triangles

Understanding triangles is a very important part of geometry. Triangles can be classified by the measures of their sides or angles. When given a specific type of triangle, you can use its special properties to solve various math problems. Having knowledge of these properties helps you set up a problem algebraically or complete a geometric proof.

The Problems You’ll Work On

In this chapter, you see a variety of geometry problems:

Identifying equilateral, isosceles, scalene, and right triangles by their side measurements

Identifying equiangular, isosceles, scalene, right, acute, and obtuse triangles by their angle measurements

Doing geometric proofs with isosceles triangles

Using the hypotenuse-leg (HL) theorem to prove triangles congruent

What to Watch Out For

Don’t let common mistakes trip you up. Some of the following suggestions may be helpful:

A triangle can be classified as more than one type of triangle. For example, a triangle can be both isosceles and right.

If you want to use HL as a method of proving triangles congruent, you must first be able to show that you have a right angle in each of the triangles.

A lot of these questions require algebraic solutions. Be sure to read the questions carefully. Sometimes a question asks you to solve for the variable, and other times the question asks for the actual measurement of the side or angle, which means you need to plug in the variable to get your solution.

Remember that it’s the degree measure of the three angles of a triangle, not the three sides of a triangle, that must add up to 180.

Remember that the congruent angles of an isosceles triangle are found opposite the congruent sides of the triangle.

Classifying Triangles by Their Sides

151–157 The given numbers represent the three sides of a triangle. Classify the triangle as isosceles, equilateral, scalene, and/or right.

151.

152.

153.

154.

155.

156.

157.

158–164 Refer to . Use the following information to calculate the length of each side of the triangle and classify the triangle as isosceles, equilateral, scalene, and/or right.

158. The perimeter of is 108 units. The three sides of the triangle are represented by

Classify this triangle.

159. The perimeter of is 210 units. The three sides of the triangle are represented by

Classify this triangle.

160. The perimeter of is 60 units. The three sides of the triangle are represented by

Classify this triangle.

161. The perimeter of is 24 units. The three sides of the triangle are represented by

Classify this triangle.

162. The perimeter of is 34 units. The three sides of the triangle are represented by

Classify this triangle.

163. The perimeter of is units. The three sides of the triangle are represented by

Classify this triangle.

164. The perimeter of is 42 units. The three sides of the triangle are represented by

Classify this triangle.

Properties of Isosceles, Equilateral, and Right Triangles

165–169 Use the properties of isosceles, equilateral, and right triangles to solve the problems.

165. is isosceles with vertex E. If and , find x.

166. is an equilateral triangle. If and , find the length of .

167. is a right triangle with . If is 3 more than and if is 3 less than two times , find the length of .

168. is isosceles with vertex H. If SH is represented by and HE is represented by , find the positive value of x.

169. is an equilateral triangle. If AL is represented by and , find the positive value of x.

Classifying Triangles by Their Angles

170–176 The given numbers represent the three angles of a triangle. Classify the triangle as acute, obtuse, equiangular, or right.

170.

171.

172.

173.

174.

175.

176.

177–182 is drawn with extended to . Use the diagram and the given information to classify each triangle as acute, obtuse, equiangular, or right.

177. and is 20 more than .

178., is represented by 2x, and is represented by .

179. is twice . Is it possible to classify this triangle?

180.

181. is 30 more than twice , and .

182.

Understanding the Classification of Triangles

183–190 Use the properties of triangles to solve the following problems.

183. is isosceles with vertex E. If and , find the degree measure of .

184. is equiangular. If is represented by and , find the value of x.

185. In right