Geometry: 1001 Practice Problems For Dummies (+ Free Online Practice)

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

In addition to what you’re reading right now, this book comes with a free, access-anywhere Cheat Sheet that includes tips and other goodies you may want to have at your fingertips. To get this Cheat Sheet, simply go to www.dummies.com and type Geometry 1001 Dummies Cheat Sheet into the Search box.

The online practice that comes free with this book offers 500 geometry questions and answers presented in a multiple-choice format. The beauty of the online problems is that you can customize your online practice to focus on the topic areas that give you trouble. If you’re short on time and want to maximize your study, you can specify the quantity of problems you want to practice, pick your topics, and go. You can practice a few hundred problems in one sitting or just a couple dozen, and whether you can focus on a few types of problems or a mix of several types. Regardless of the combination you create, the online program keeps track of the questions you get right and wrong so you can monitor your progress and spend time studying exactly what you need.

To gain access to the online practice, you simply have to register. Just follow these steps:

Register your book or ebook at Dummies.com to get your PIN. Go towww.dummies.com/go/getaccess.

Select your product from the dropdown list on that page.

Follow the prompts to validate your product, and then check your email for a confirmation message that includes your PIN and instructions for logging in.

If you don’t receive this email within two hours, please check your spam folder before contacting us through our Technical Support website at http://support.wiley.com or by phone at 877-762-2974.

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

Your registration is good for one year from the day you activate your PIN.

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 1

The Questions

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 math , then math .

2. If math bisects math at E, then math .

3. If math , then _______ is a right angle.

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

4. math is a right angle.

5. math .

6. math and math form a linear pair.

7. math .

8. math is an obtuse angle.

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

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

10. math is the median of math .

11. math is the altitude of math .

12. math bisects math .

13. F is the midpoint of math .

14. F is the midpoint of math . What type of angle does math have to be in order for math 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 math . If math and math , find the value of x.

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

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

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

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 math to fill in the blank.

27. math

28. math

29. math

30. math

31. The ___________________ would be the reason used to prove that math .

32. If math , then math .

33. If math , then math .

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

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

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

36. math

37. math

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

39. If math bisects math , you can conclude that math .

40. If math bisects math , you can conclude that math .

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

41–47 In the following figure, math intersects math at E. Fill in the blank to make the statement true.

41. math because they’re vertical angles.

42. math and math are ___________________ angles.

43. math and math form a linear pair; therefore, the two angles add up to ___________________.

44. math is represented by math , and math is represented by math . math ?

45. math , math , and math are represented by math , math , and math , respectively. math ?

46. math and math are angles that share the same vertex and are next to each other. These are called ___________________ angles.

47. math and math form a linear pair. If math is represented by math and math is represented by math , then what does math equal?

Complementary and Supplementary Angles

48–57 Practice understanding angle relationships by solving the problem algebraically.

48. math and math are complementary. If math , find math .

49. math and math are supplementary. If math , find math .

50. If two angles are complementary and congruent to each other, what is the measure of the angles?

51. Two angles are supplementary and congruent. What type of angles must they be?

52. The ratio of two angles that are supplements is 2:3. Find the larger angle.

53. If two angles are supplementary and one angle is 40° more than the other angle, find the smaller angle.

54. If two angles are complementary and one angle is twice the measure of the other, find the measure of the smaller angle.

55. If two angles are complementary and one angle is 6 less than twice the measure of the other angle, find the larger angle.

56. If two angles form a linear pair, what is their sum?

57. The ratio of two angles that are complements of each other is 5:4. Find the measure of the smaller angle.

Angles in a Triangle

58–60 Use the following figure and the given information to solve each problem. math and math intersect at E.

58. If math , math , and math , find the value of x.

59. If math and math , find the degree measure of math .

60. If math is represented by math , math is represented by math , and math is represented by math , find the value of a.

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 30°-60°-90° and 45°-45°-90° 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 math , a line segment congruent to math .

62. Construct math an angle congruent to math

63. Construct math , a triangle congruent to math .

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

65. Construct math , a triangle congruent to math .

Constructions Involving Angles and Segments

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

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

67. Given math , construct math , the bisector of math .

68. Construct the angle bisector of math .

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

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

Parallel and Perpendicular Lines

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

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

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

73. Construct the perpendicular bisector of math .

74. Which construction is represented in the following figure?

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

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

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

Creative Constructions

78–85 Apply your knowledge of constructions to some more creative problems.

78. Construct a math angle.

79. True or False? The following diagram shows the first step in constructing a math angle.

80. Construct an altitude from vertex A to side math in math .

81. Construct the median to math in math .

82. Construct an equilateral triangle whose side length is math .

83. Construct a math angle.

84. Construct a math triangle.

85. Construct the median to math in math .

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.

Triangle Congruence Theorems

86–102 Use your knowledge of SAS, ASA, SSS, and AAS to solve the problem.

86. What method can you use to prove these two triangles congruent?

87. What method can you use to prove these two triangles congruent?

88. What method can you use to prove these two triangles congruent?

89. What method can you use to prove math

90. Which pair of segments or angles would need to be proved congruent in order to prove these triangles congruent using the SSS method?

91. Which pair of segments or angles would need to be proved congruent in order to prove these triangles congruent using the SAS method?

92. Which pair of segments or angles would need to be proved congruent in order to prove these triangles congruent using the AAS method?

93. Which pair of segments or angles would need to be proved congruent in order to prove these triangles congruent using the ASA method?

94. Which pair of segments or angles would need to be proved congruent in order to prove these triangles congruent using the SSS method?

95. Which pair of segments or angles would need to be proved congruent in order to prove the triangles congruent using the SAS method?

96. Given: math bisects math and E is the midpoint of math . Is it possible to prove math using only the given information and the reflexive property?

97. Given: math bisects math and math bisects math . Which method of triangle congruence would you use to prove math

98. Given: math and math . Which method of triangle congruence would you use to prove math

99. Given: math is the altitude drawn to math , and math bisects math . Which method of triangle congruence would you use to prove math

100. Given: math and math . Which method of triangle congruence would you use to prove math

101. Given: math , and math . Which method of triangle congruence would you use to prove math

102. Given: Quadrilateral math , and math . Which method of triangle congruence would you use to prove math

Completing Geometric Proofs Using Triangle Congruence Theorems

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

Given: math and math bisect each other at B.

Prove: math

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