Daniel C. Alexander

628 Pages·2010·11.73 MB·English

by  Daniel C. Alexander

Formulas
PLANE FIGURES: Kite:
P(cid:2) Perimeter; C(cid:2) Circumference; A(cid:2)Area a b P = 2a + 2b
d1 A = 1 #d #d
Triangle: a d2 b 2 1 2
P (cid:2) a (cid:3) b (cid:3) c
c 1
a h A = bh Circle:
2
A = 2s(s - a)(s - b)(s - c), C = 2pr or C = pd
b r A = pr2
wheres(cid:2)semiperimeter
Equilateral Triangle:
P(cid:2) 3s
A = s4223 Regular Polygon P(n=sidnes#)s:
1
s A = aP
s 2
a
Rectangle:
P = 2b + 2h
MISCELLANEOUS FORMULAS:
A(cid:2)bhor A(cid:2)(cid:2)w
h
Right Triangle:
b c2 = a2 + b2
1
c b A = ab
Parallelogram: 2
P = 2a + 2b a
A = bh
a h
Polygons (nsides): #
b Sum (interior angles) (cid:2)(n - 2) 180°
Sum (exterior angles) (cid:2)360°
n(n - 3)
Trapezoid: Number (of diagonals) (cid:2)
b1 P = a + b1 + c + b2 2
1
A = h(b + b )
a h c 2 1 2 Regular Polygon (nsides): I(cid:2)measure Interior angle,
E(cid:2)measure Exterior angle,#and C(cid:2)measure Central angle
b2 I = (n - 2) 180°
E n
I
360°
Square: E(cid:2)
P(cid:2)4s C n
A(cid:2)s2 C = 360°
n
s
Sector:
¬
¬ mAB
Rhombus: A /AB = * 2pr
P(cid:2)4s ¬360°
s dd21 A = 21 # d1 #d2 B A = m36A0B° * pr2
s
SOLIDS (SPACE FIGURES): ANALYTIC GEOMETRY:
L(cid:2) Lateral Area; T(or S) (cid:2)Total (Surface)
Area; V(cid:2)Volume y Distance:
d(cid:2)(cid:2)(x (cid:5)x )2(cid:3)(y (cid:5)y )2
10 2 1 2 1
Parallelepiped (box)h: VT == 2//wwh + 2/h + 2wh (x1, y1) 8642 (x2, y2) MMid=poainxt1:+2 x2, y1 +2 y2b
w –10–8–6–4–2––24 2 4 6 8 10 x Slope: m = xy2 -- yx1, x1 Z x2
2 1
–6
Parallel Lines:
–8
–10 / ||/ 4m = m
1 2 1 2
Right Prism: Perpendicular L# ines:
L = hP / (cid:2) / 4m m = - 1
1 2 1 2
T = L + 2B
h V = Bh EquSaltoiopnes-I onfte ar cLeipnte::y = mx + b
Point-Slope: y - y = m(x - x )
1 1
General: Ax + By = C
Regular Pyramid:
1
L = /P TRIGONOMETRY:
2
h
/2 = a2 + h2
a T = L + B Right Triangle:
1 opposite a
V = Bh sin u = =
3 c a hypotenuse c
adjacent b
(cid:4) cos u = =
b hypotenuse c
Right Circular Cylinder:
L = 2prh tan u = opposite = a
T = 2prh + 2pr2 adjacent b
h V = pr2h sin2 u + cos2 u = 1
r
Triangle:
Right Circular Cone:
L = pr/ (cid:6) A = 1bc sin a
/2 = r2 + h2 b a 2
h sin a sin b sin g
T = pr/ +pr 2 (cid:7) (cid:8) = =
c a b c
r V = 1pr2h c2 = a2 + b2 - 2ab cos g
3
or
a2 + b2 - c2
cos (cid:2) =
Sphere: 2ab
S = 4pr2
r
4
V = pr3
3
Miscellaneous:
Euler’s Equation: V(cid:3)F(cid:2)E(cid:3)2
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Fifth Edition
Elementary Geometry
for College Students
Daniel C. Alexander
Parkland College
Geralyn M. Koeberlein
Mahomet-Seymour High School
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Elementary Geometry for College Students, © 2011, 2007 Brooks/Cole, Cengage Learning
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This edition is dedicated to our spouses, children, and grandchildren.
Dan Alexander and Geralyn Koeberlein
LETTER FROM THE AUTHOR
Through many years of teaching mathematics, particularly geometry, I found that
geometry textbooks were lacking—lacking “whats, whys, and how tos.”As I taught
this subject,I amassed huge piles of notes that I used to supplement the text in class
discussions and lectures. Because some explanations were so lacking in the textbooks,
I found myself researching geometry to discover new and improved techniques,alter-
native approaches,and additional proofs and explanations. When unable to find what
I sought,I often developed a more concise or more easily understood explanation of
my own.
To contrast the presentation of geometry with a sportscast,geometry textbooks of-
ten appeared to me to provide the play-by-play without the color commentary. I found
that entire topics might be missing and figures that would enable the student to “see”
results intuitively were not always provided. The explanation of why a theorem must
be true might be profoundly confusing,unnecessarily lengthy,or missing from the text-
book altogether. Many geometry textbooks avoided proof and explanation as if they
were a virus. Others would include proof,but not provide any suggestions or insights
into the synthesis of proof.
During my years teaching at Parkland College,I was asked in the early 1980s to
serve on the geometry textbook selection committee. Following the selection, I dis-
covered serious flaws as I taught from the “best” textbook available. Really very
shocking to me—I found that the textbook in use contained errors,including errors in
logic that led to contradictions and even to more than one permissible answer for some
problems.
At some point in the late 1980s,I began to envision a future for the compilation of
my own notes and sample problems. There was,of course,the need for an outline of
the textbook to be certain that it included all topics from elementary geometry. The
textbook would have to be logical to provide a “stepping stone”approach for students.
It would be developed so that it paved the way with explanation and proofs that could
be read and understood and would provide enough guidance that a student could learn
the vocabulary of geometry,recognize relationships visually,solve problems,and even
create some proofs. Figures would be included if they provided an obvious relationship
where an overly wordy statement of fact would be obscure. The textbook would have
to provide many exercises,building blocks that in practice would transition the student
from lower level to mid-range skills and also to more challenging problems.
In writing this textbook for college students,I have incorporated my philosophy
for teaching geometry. With each edition, I have sought to improve upon an earlier
form. I firmly believe that the student who is willing to study geometry as presented
here will be well prepared for future study and will have developed skills of logic that
are enduring and far-reaching.
Daniel C. Alexander
iv
Contents
Preface ix
Foreword xvii
Index of Applications xviii
1
Line and Angle Relationships 1
1.1 Sets, Statements, and Reasoning 2 (cid:2) PERSPECTIVE ON HISTORY:The Development
ofGeometry 60
1.2 Informal Geometry and Measurement 10
(cid:2) PERSPECTIVE ON APPLICATION: Patterns 60
1.3 Early Definitions and Postulates 21
SUMMARY 62
1.4 Angles and Their Relationships 30
REVIEW EXERCISES 65
1.5 Introduction to Geometric Proof 39
CHAPTER 1 TEST 68
1.6 Relationships: Perpendicular Lines 46
1.7 The Formal Proof of a Theorem 53
2
Parallel Lines 71
2.1 The Parallel Postulate and Special Angles 72 (cid:2) PERSPECTIVE ON HISTORY: Sketch of Euclid 118
2.2 Indirect Proof 80 (cid:2) PERSPECTIVE ON APPLICATION: Non-Euclidean
2.3 Proving Lines Parallel 86 Geometries 118
2.4 The Angles of a Triangle 92 SUMMARY 120
2.5 Convex Polygons 99 REVIEW EXERCISES 123
2.6 Symmetry and Transformations 107 CHAPTER 2 TEST 125
3
Triangles 127
3.1 Congruent Triangles 128 (cid:2) PERSPECTIVE ON HISTORY: Sketch of
3.2 Corresponding Parts of Congruent Archimedes 168
Triangles 138 (cid:2) PERSPECTIVE ON APPLICATION:
3.3 Isosceles Triangles 145 Pascal’s Triangle 168
3.4 Basic Constructions Justified 154 SUMMARY 170
3.5 Inequalities in a Triangle 159 REVIEW EXERCISES 172
CHAPTER 3 TEST 174
v
vi CONTENTS
4
Quadrilaterals 177
4.1 Properties of a Parallelogram 178 (cid:2) PERSPECTIVE ON APPLICATION: Square Numbers as
4.2 The Parallelogram and Kite 187 Sums 211
4.3 The Rectangle, Square, and Rhombus 195 SUMMARY 212
4.4 The Trapezoid 204 REVIEW EXERCISES 214
(cid:2) PERSPECTIVE ON HISTORY: Sketch of Thales 211 CHAPTER 4 TEST 216
5
Similar Triangles 219
5.1 Ratios, Rates, and Proportions 220 (cid:2) PERSPECTIVE ON HISTORY: Ceva’s Proof 269
5.2 Similar Polygons 227 (cid:2) PERSPECTIVE ON APPLICATION: An Unusual
5.3 Proving Triangles Similar 235 Application of Similar Triangles 269
5.4 The Pythagorean Theorem 244 SUMMARY 270
5.5 Special Right Triangles 252 REVIEW EXERCISES 273
5.6 Segments Divided Proportionally 259 CHAPTER 5 TEST 275
6
Circles 277
6.1 Circles and Related Segments (cid:2) PERSPECTIVE ON HISTORY: Circumference
and Angles 278 oftheEarth 316
6.2 More Angle Measures in the Circle 288 (cid:2) PERSPECTIVE ON APPLICATION: Sum of the Interior
6.3 Line and Segment Relationships in Angles of a Polygon 316
the Circle 299 SUMMARY 317
6.4 Some Constructions and Inequalities for REVIEW EXERCISES 319
theCircle 309
CHAPTER 6 TEST 321
7
Locus and Concurrence 323
7.1 Locus of Points 324 (cid:2) PERSPECTIVE ON APPLICATION: The Nine-Point Circle
7.2 Concurrence of Lines 330 346
7.3 More About Regular Polygons 338 SUMMARY 347
(cid:2) PERSPECTIVE ON HISTORY: The Value of (cid:2) 345 REVIEW EXERCISES 349
CHAPTER 7 TEST 350
8
Areas of Polygons and Circles 351
8.1 Area and Initial Postulates 352 (cid:2) PERSPECTIVE ON APPLICATION: Another Look at the
8.2 Perimeter and Area of Polygons 363 Pythagorean Theorem 394
8.3 Regular Polygons and Area 373 SUMMARY 396
8.4 Circumference and Area of a Circle 379 REVIEW EXERCISES 398
8.5 More Area Relationships in the Circle 387 CHAPTER 8 TEST 400
(cid:2) PERSPECTIVE ON HISTORY:
Sketch of Pythagoras 394