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Geometry
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6380-7 Cover 4/18/01 12:22 PM Page 1
CliffsQuickReview™
Geometry
By Ed Kohn, MS
An International Data Group Company
New York, NY • Cleveland, OH • Indianapolis, IN
6380-7 FM.F 4/30/01 9:22 AM Page i
CliffsQuickReview™Geometry
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6380-7 FM.F 4/30/01 9:22 AM Page ii
Table of Contents
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1
Why You Need This Book . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1
How to Use This Book . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2
Don’t Miss Our Web Site . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2
Chapter 1: Fundamental Ideas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3
Points, Lines, and Planes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3
Point . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .4
Line . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .4
Plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .5
Postulates and Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .5
Segments, Midpoints, and Rays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .7
Line segment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .7
Midpoint . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .9
Ray . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .9
Angles and Angle Pairs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .10
Angles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .10
Angle bisector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .13
Right angle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .13
Acute angle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .14
Obtuse angle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .14
Straight angle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .15
Special Angles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .15
Adjacent angles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .16
Vertical angles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .16
Complementary angles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .17
Supplementary angles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .18
Intersecting, Perpendicular, and Parallel Lines . . . . . . . . . . . . . . . . . .19
Intersecting lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .19
Perpendicular lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .19
Parallel lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .20
Parallel and Perpendicular Planes . . . . . . . . . . . . . . . . . . . . . . . . . . . .20
Parallel planes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .20
Perpendicular planes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .21
Chapter 2: Parallel Lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .24
Angle Pairs Created with a Transversal . . . . . . . . . . . . . . . . . . . . . . . .24
The Parallel Postulate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .26
Consequences of the Parallel Postulate . . . . . . . . . . . . . . . . . . . . . . . .26
Testing for Parallel Lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .28
6380-7 FM.F 4/30/01 9:22 AM Page iii
Chapter 3: Triangles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .34
Angle Sum of a Triangle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .35
Exterior Angle of a Triangle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .35
Classifying Triangles by Sides or Angles . . . . . . . . . . . . . . . . . . . . . . .36
Special Names for Sides and Angles . . . . . . . . . . . . . . . . . . . . . . . . . .38
Altitudes, Medians, and Angle Bisectors . . . . . . . . . . . . . . . . . . . . . . .39
Base and altitude . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .39
Median . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .41
Angle bisector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .41
Congruent Triangles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .43
Corresponding parts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .43
Tests for congruence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .44
Special Features of Isosceles Triangles . . . . . . . . . . . . . . . . . . . . . . . . .50
Triangle Inequalities Regarding Sides and Angles . . . . . . . . . . . . . . . .52
The Triangle Inequality Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . .53
Chapter 4: Polygons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .55
Classifying Polygons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .55
Identifying the parts of a polygon . . . . . . . . . . . . . . . . . . . . . . . . .56
Number of sides . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .57
Regular polygons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .58
Angle Sum of Polygons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .58
Special Quadrilaterals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .60
Trapezoid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .61
Parallelogram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .61
Proving That Figures Are Parallelograms . . . . . . . . . . . . . . . . . . . . . .63
Properties of Special Parallelograms . . . . . . . . . . . . . . . . . . . . . . . . . .64
Rectangle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .64
Rhombus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .64
Square . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .65
Properties of Trapezoids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .67
The Midpoint Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .70
Chapter 5: Perimeter and Area . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .72
Squares and Rectangles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .72
Finding the perimeter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .72
Finding the area . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .73
Parallelograms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .74
Finding the perimeter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .75
Finding the area . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .75
Triangles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .75
Finding the area . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .76
Finding the perimeter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .76
iv CliffsQuickReview Geometry
6380-7 FM.F 4/30/01 9:22 AM Page iv
Trapezoids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .77
Finding the perimeter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .77
Finding the area . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .78
Regular Polygons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .79
Parts of a regular polygon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .79
Finding the perimeter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .79
Finding the area . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .79
Circles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .80
Finding the circumference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .80
Finding the area . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .81
Summary of Perimeter, Circumference, and Area Formulas . . . . . . . .82
Chapter 6: Similarity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .84
Ratio and Proportion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .84
Ratio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .84
Proportion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .86
Means and extremes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .86
Properties of Proportions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .86
Similar Polygons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .89
Similar Triangles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .90
Proportional Parts of Triangles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .93
Proportional Parts of Similar Triangles . . . . . . . . . . . . . . . . . . . . . . . .95
Perimeters and Areas of Similar Triangles . . . . . . . . . . . . . . . . . . . . . .97
Chapter 7: Right Triangles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .102
Geometric Mean . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .102
Altitude to the Hypotenuse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .103
Pythagorean Theorem and Its Converse . . . . . . . . . . . . . . . . . . . . . .106
Extension to the Pythagorean Theorem . . . . . . . . . . . . . . . . . . . . . .110
Special Right Triangles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .112
Chapter 8: Circles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .118
Parts of Circles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .118
Central Angles and Arcs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .121
Central angles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .121
Arcs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .122
Arcs and Inscribed Angles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .126
Other Angles Formed by Chords, Secants, and Tangents . . . . . . . . .130
Arcs and Chords . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .135
Segments of Chords, Secants, and Tangents . . . . . . . . . . . . . . . . . . .138
Arc Length and Sectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .143
Arc length . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .143
Sector of a circle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .144
Summary of Angle, Segment, Arc Length,
and Sector Relationships . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .146
Table of Contents v
6380-7 FM.F 4/30/01 9:22 AM Page v
Chapter 9: Geometric Solids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .149
Prisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .149
Right Prisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .150
Lateral area of a right prism . . . . . . . . . . . . . . . . . . . . . . . . . . . . .150
Total area of a right prism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .151
Interior space of a solid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .152
Right Circular Cylinders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .154
Pyramids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .156
Regular Pyramids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .156
Right Circular Cones . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .159
Spheres . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .161
Chapter 10: Coordinate Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . .164
Points and Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .164
Distance Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .167
Midpoint Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .168
Slope of a Line . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .170
Slopes of Parallel and Perpendicular Lines . . . . . . . . . . . . . . . . . . . .172
Equations of Lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .173
Summary of Coordinate Geometry Formulas . . . . . . . . . . . . . . . . . .178
CQR Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .180
CQR Resource Center . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .188
Glossary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .190
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .196
vi CliffsQuickReview Geometry
6380-7 FM.F 4/30/01 9:22 AM Page vi
Introduction
Learning geometry has traditionally been regarded as important in the sec-
ondary schools, at least partly because it has been the primary means of
teaching the art of reasoning. Over time, geometry has evolved into a beau-
tifully arranged and logically organized body of knowledge. It consists pri-
marily of a sequence of statements about points, lines, and planes, as well as
planar and solid figures. Geometry begins with undefined terms, definitions,
and assumptions; these lead to theorems and constructions. It is an abstract
subject, but easy to visualize, and it has many concrete practical applications.
Historically, geometry has long been important for its role in the surveying
of land and more recently, our knowledge of geometry has been applied to
help build structurally sound bridges, experimental space stations, and large
athletic and entertainment arenas, just to mention a few examples.
Unfortunately, many students planning to take standardized college admis-
sion tests or to pursue careers requiring more advanced mathematics and
physics overlook the significant role of geometry in these undertakings.
This book helps you get ready for success on the mathematics portion of
these standardized tests or in the more advanced mathematics and science
courses required for careers in mathematics, the sciences, and engineering.
Teachers and students alike will find CQR Geometry to be a valuable course
supplement.
The prerequisites for subject comprehension include familiarity with basic
arithmetic operations and with some topics from introductory algebra.
The arithmetic prerequisites include converting fractions to decimals, con-
verting improper fractions to mixed numbers, and taking square roots.
The algebraic prerequisites include addition of equations, solving linear
equations in two variables for one variable in terms of the other, and fac-
toring quadratic equations.
Why You Need This Book
Can you answer yes to any of these questions?
■Do you need to review the fundamentals of geometry fast?
■Do you need a course supplement to geometry?
■Do you need to prepare for your geometry test?
■Do you need a concise, comprehensive reference for geometry?
6380-7 Intro.F 4/30/01 9:22 AM Page 1
If so, then Cliffs Quick Review Geometry is for you!
How to Use This Book
Because mathematics builds on itself, many readers benefit most from
studying or reviewing this book from cover to cover. However, you’re the
boss here, and you may choose to seek only the information you want and
then put the book back on the shelf for later. In that case, here are a few
recommended ways to search for topics:
■Flip through the book, looking for your topic in the running heads.
■Look in the Glossary for all the important terms and definitions.
■Look for your topic in the Table of Contents in the front of the book.
■Look at the In This Chapter section at the front of each chapter.
■Look at the review questions in the Chapter Checkout.
■Look for additional information in the CQR Resource Center or test
your knowledge with the CQR Review.
■Refer to the CQR Pocket Guide.
■Flip through the book and look at the many figures, reading the cap-
tions until you find what you’re looking for.
Don’t Miss Our Web Site
Keep up with the latest resources available by visiting the CliffsNotes web
site at www.cliffsnotes.com. Here’s what you find:
■Informative interactive tools
■Links to related web sites
■Additional resources to help you continue your learning
At www.cliffsnotes.com, you can even register for a new feature called
CliffsNotes Daily, which offers you newsletters on a variety of topics and
is delivered right to your e-mail inbox each business day.
If you haven’t yet discovered the Internet and are wondering how to get
online, pick up Getting On the Internet, new from CliffsNotes. You’ll learn
just what you need to make your online connection quickly and easily. See
you at www.cliffsnotes.com!
2CliffsQuickReview Geometry
6380-7 Intro.F 4/30/01 9:22 AM Page 2
摘要：

GeometryGeometryTheessentialsfromtheexpertsatCliffsNote-A-Day™tipsatcliffsnotes.comMasterthebasics—fastCompletecoverageofcoreconceptsAccessible,topic-by-topicorganizationFreepocketguideforeasyreferenceWetakegreatnotes—andmakelearningasnapWhenitcomestopinpointingthestuffyoureallyneedtoknow,nobodydoes...

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