Integrated Mathematics

721 Pages·1990·83.7 MB·English

by  Dressler I.

lntegrated
Mathematrcs
II
couRse
Second Edition
AUTHORS
Edward P. Keenan
lsidore Dressler REVISERS
Ann Xavier Gantert
Marilyn 0cchiogrosso
our nalion's youth
When ordering this book, please specify: either R 514 H or
INTEGRATED MATHEMATICS: COURSE II, SECOND EDITION, HARDBOUND
AMSCO SCHOOL PUBLICATIONS, INC.
Street
315 Hudson New York, N.Y. 10013
Edward P. Keenan
C urriculurn A ssociate, Mathematics
East Williston Union Free School District
East Williston, New York
lsidore Dressler
Former Chairrnan
D epartment of Mathematics
Bayside High School, New York City
Ann Xavier Gantert
Departrnent of Mathematics
Nazareth Academy
Rochester, New York
Marilyn 0cchiogrosso
Former Assistant Principal
Mathematics
Erasmus Hall High School, New York City
ISBN 0-87720-272-9
Copyright O 1990, 1981 by Amsco School Publications, Inc.
No part of this book may be reproduced in any form
without written permission from the publisher.
PRINTED IN THE UNITED STATES OF AMERICA
Preface
INrpcRarso MerHorvrerlcs: CouRsn lI, Second Edition, is a thorough
revision of a textbook that has been a leader in presenting high school
mathematics in a contemporary, integrated manner. Over the last
decade, this integrated approach has undergone further changes and
refinements. Amsco's Second Edition reflects these developments.
The Amsco book parallels the integrated approach to the teaChing of
high school mathematics that is being promoted by the National
Council of Teachers of Mathematics (NCTM) in its STANDARDS FOR
SCHOOL MATHEMATICS. Moreover, the Amsco book implements
many of the suggestions set forth in the NCTM Standards, which are
the acknowledged guidelines for achieving a higher level of excellence
in the study of mathematics.
In this new edition, which fully satisfies the requirements of the
revised New York State Syllabus:
o
Problem solving has been expanded by (1) adding nonroutine prob-
lems for selected topics and to Chapter Reviews, and (2) providing, in the
Teacher's Manual, Bonus questions for each chapter.
o Integration of Geometry, Logic, Algebra, and other branches of
mathematics, for which the First Edition was well known, has been
broadened by the inclusion of new topics and the earlier introduction of
selected concepts.
o
Algebraic skills from Course I have been maintained, strengthened,
and expanded as a bridge to the requirements of Course III. Note that
many of these skills, newly highlighted in the revised Syllabus, already
appear in the First Edition of the AMSCO text.
o Enrichment has been extended by (1) increasing the number of
challenging exercises, (2) introducing a variety of optional topics, and
(3) adding to the Teacher's Manual more thought-provoking aspects of
topics in the text, and supplementary material that reflects current
thinking in mathematics education.
o Hands-on activities have been included in the Teacher's Manual to
promote understanding through discovery.
iii
lv lntegrated Mathematics: Course ll
The First Edition of the text had been written to provide effective
teaching materials for a unified program appropriate for 10th-grade
mathematics students, including topics not previously contained in a
traditional Geometry course. These topics-Logic, Probability, Mathe-
matical Systems, Transformation Geometry, and Quadratic-Linear
Systems-are retained in the Second Edition. In addition, some have been
expanded.
While Course I of this series is concerned with an intuitiue approach
to mathematics, the keystone of Course II is proof.
In this text, geometry is developed as a postulational system ofreason-
ing, beginning with a review of definitions in Chapter 1. As the need for
proof develops, Chapter 2 is devoted to a study of the laws of reasoning
and logic proofs. A unique blending occurs in Chapter 3, wherein students
are shown how proofs in logic are related to traditional deductive proofs
in geometry, both direct and indirect. The integration of logic and geom-
etry is seen throughout the text in the formulation of definitions and in
many proofs, most notably those in Locus. Mathematical Systems, intro-
'duced in Chapter 14, provides the student with a completely different
postulational system ofreasoning, one based on arithmetic and algebra
rather than on geometry, but leading to proofby the end ofthe chapter.
An intent of the authors was to make the original book of greatest
service to average students. Since its publication, however, the text has
been used successfully with students of varying ability levels. To main-
tain this broad spectrum of use, the basic elements of the original work
have been preserved in the Second Edition. Once again:
o
Concepts are carefully developed, using appropriate rigor and math-
ematical symbolism.
a
Definitions, postulates, theorems, corollaries, principles, and proce-
dures are stated precisely and explained by specific examples.
o
The numerous model problems are solved through detailed step-by-
step explanations.
o
Varied and carefully graded exercises are provided in abundance.
This new edition is offered so that teachers may effectively continue
to help students comprehend, master, and enjoy mathematics from an
integrated point of view.
The Authors
a personal note
As the first edition of Integrated Mathematics: Course 11 was being
written in 1980, Isidore Dressler became seriously ill. Before the book
was completed that year, he died.
For many years, Isidore Dressler influenced teachers and students
alike. Hundreds of thousands of students used his numerous books and
were helped to learn mathematics by the style and clarity of his expo-
sition and exercises. Portions of his Geornetry have been used in both
editions of Integrated Mathematics: Course //. For this, many people will
be grateful.
It was a rare privilege for me to work as a coauthor with Isidore
Dressler. As an educator and an author, Isidore Dressler cared about
two mythical students he called "Average Joe" and "Average Jane." He
taught and wrote with them always in mind. His books will live on, help-
ing many more students learn mathematics. A warm, loving human
being, he also cared deeply about his family, for whom the memory of a
very special man will live on.
Faced with the task of organizing and writing portions of this book
alone, I shall always be very grateful for the valuable criticisms and
suggestions offered by Sister Ann Xavier Gantert of Nazareth Academy,
Rochester, New York.
This book is dedicated to the memory of Isidore Dressler, and to Anna
Keenan, whose perseverance, strength, and love influenced her son to
pursue a higher education and higher goals than he once had. This
author is forever indebted to both of these people.
Edward P. Keenan
v

Contents
1
CHAPTER lntroducing Geometry
I -1 The Meaning of Geometry 1
l-2
Properties of the Real Numbers 2
1-B
Undefined Terms 4
l-4 Definitions Involving Lines and Line Segments 5
1-5 Definitions Involving Angles t4
.
1-6
Definitions Involving Pairs of Angles . 20
l-7 Definitions Involving Perpendicular Lines 25
1-8
Definitions Involving Triangles and
Line Segments Associated With Tliangles 27
1-9
Reuiew Exercises 31
2
CHAPTER Logic
2-l
The Need for Logic 34
2-2
Sentences,Statements, and TYuth Values . 35
2-g Connectives in Logic 35
2-4 Tluth Tables, Tautologies, and
Statements
Logically Equivalent 43
2-5 Detachment
The Law of 47
2-G The Law of the Contrapositive; Proof in Logic 55
2-7 .
The Law of Modus Tollens 61
2-8 Invalid Arguments 66
2-g .
The Chain Rule 7L
2-10 The Law of Disjunctive Inference 78
2-11 Negations and De Morgan's Laws 83
2-12 The Laws of Simplification, Conjunction, and
Disjunctive Addition 88
2-13 Practice With Logic Proofs 91
.
2-14 Reuiew Exercises 95
CHAPTER 3 Proving Statements in Geometry
3-1 .
Inductive Reasoning 98
3-2 Definitions as Biconditionals . 100
3-3 Reasoning
Deductive 103
vil
vlll lntegrated Mathematlcs: Course ll
3-4 Direct Proof and Indirect Proof t]-z
3-5 Understanding the Nature of a Postulational System tt7
3-6 The First Postulates Used in Proving Conclusions . 119
3-7 Equivalence Relations L22
3-8 More Postulates and Proofs t24
3-9 Using Postulates and Definitions in Proofs 139
3-10 Proving Simple Angle Theorems 142
3-17 Reuiew Exercises 148
CHAPTER 4 Triangle Congruence and lnequalities
4-L Congruent Polygons and Corresponding Parts 150
.
4-Z
Proving Tliangles Congruent When Two Pairs of Sides
and the Included Angle Are Congruent 154
.
4-g
Proving Triangles Congruent When T\vo Pairs of Angles
and the Included Side Are Congruent 158
4-4
Proving TYiangles Congruent When Three Pairs of Sides
Congruent
Are 160
4-5 More Line Segments Associated With Tliangles . L62
4-6 More Practice in Proving Tliangles Congruent 164
4-7 Using Congruent Tliangles to Prove Line Segments
Congruent and Angles Congruent 168
4-8 The Isosceles Tliangle and the Equilateral Tliangle . L72
4-g Triangles
Using Two Pairs of Congruent 175
4-10 Proving Overlapping Tliangles Congruent 178
4-11 Using Basic Inequality Postulates 181
4-12 Inequality Postulates Involving Operations 185
4-13 An Inequality Involving the Lengths of the Sides
of a Tliangle 190
4-L4 An Inequality Involving an Exterior Angle of a Tliangle r92
4-15 Inequalities Involving Sides and Angles in a Ttiangle 198
4-LG Reuiew Exercises 203
CHAPTER 5 Perpendicular and Parallel Lines, Angle Sums,
and More Congruences
5-1
Proving Lines Perpendicular . 207
5-2 Proving Lines Parallel 270
Contents tx
.
5-3 Converse Statements; Properties of Parallel Lines 217
T?iangle
5-4 The Sum of the Measures of the Angles of a 223
5-5 Proving Tliangles Congruent When T\vo Pairs of Angles
and a Pair of Opposite Sides Are Congruent 227
5-G The Converse of the Isosceles TYiangle Theorem 23L
5-7 Proving Right TYiangles Congruent by
Hypotenuse and Leg 235
5-8 Exterior Angles of a Triangle 238
5-g Interior Angles and Exterior Angles of a Polygon 24t
5-lO Reuiew Exercises 244
CHAPTER6 Quadrilaterals
6-1 The General Quadrilateral . 247
6-2 The Parallelogram 247
6-3 The Rectangle 254
6-4 The Rhombus 258
.
6-5 The Square 263
6-6 The Tlapezoid 265
6-7 Reuiew Exercises 269
CHAPTER 7 Similarity; Special Triangles
7-l Square Roots and Radicals . 273
7-2 Ratio and Proportion 277
7-g Ratio, Proportion, and Congruent Line Segments 282
7-4 Proportions Involving Line Segments ' 287
7-5 Similar Polygons 290
7-G Using Similar Tliangles to Prove Proportions Involving
Line Segments 301
7-7 Proving That Products Involving Line
Segments Are Equal 303
7-8 Using Proportions Involving Corresponding Line
Segments in Similar Tiiangles . 305
7-9 Proportions in the Right Tliangle 309
7-10 The Pythagorean Theorem and Its Applications 3t4
7-11 Special Bight TYiangles 322
7-12 Reuiew Exercises 329