Geometry, Its Elements and Structure: Second Edition

826 Pages·2014·25.31 MB·English

by  Alfred S. Posamentier

GEOMETRY
Its Elements & Structure
SECOND EDITION
Alfred S. Posamentier
Robert L. Bannister
DOVER PUBLICATIONS, INC.
Mineola, New York
Readers of this book who would like to receive a Solutions Manual may request
it from the publisher at the following e-mail address:
[email protected]
Copyright
Copyright © 1972, 1977, 2014 by Alfred S. Posamentier and Robert L.
Bannister
Introduction to the Dover Edition Copyright © 2014
by Alfred S. Posamentier
All rights reserved.
Bibliographical Note
This Dover edition, first published in 2014, is a corrected, revised republication
of the 1977 second edition of the work originally published in 1972 by the
McGraw-Hill Book Company, New York. Alfred S. Posamentier has provided a
new Introduction to this Dover edition.
Library of Congress Cataloging-in-Publication Data
Posamentier, Alfred S. Geometry, its elements and structure / Alfred S.
Posamentier, Robert L. Bannister.
pages cm
“This Dover edition, first published in 2014, is an unabridged republication of
the 1977 second edition of the work originally published in 1972 by the
McGraw-Hill Book Company, New York”–Preliminaries.
Includes index.
Summary: “Co-written by a bestselling high school and university textbook
author, a longtime educational and standards pioneer, this up-to-date text is
geared toward high school geometry classes and contains standard material for
numerous state competencies. Topics include plane, solid, coordinate, vector,
and non-Euclidean geometry. Features more than 2,000 illustrations, numerous
examples with worked-out solutions, and supplementary reading.”
– Provided by publisher.
eISBN-13: 978-0-486-78216-4
1. Geometry. I. Bannister, Robert L. II. Title.
QA453.B36 2014
516.2-dc23 2013016251
Manufactured in the United States by Courier Corporation
49267201 2014
www.doverpublications.com
Introduction to the Dover Edition
The time-honored geometry course typically taught in high schools over the last
hundred years has a long history and is clearly unique to the United States.
Today it is manifested within the Common Core State Standards. Although most
believe that this course stems largely from Euclid’s elements, it would probably
be more accurate to say that the geometry book published in English by the
Scottish mathematician Robert Simson (1687–1768), and which remained in
print for well over one hundred years, had probably the greatest influence on our
one-year-long high school study of geometry. He supported the Euclidean style
and shied away very directly from that of Rene Descartes (1596–1650). Perhaps
one of the greatest influences for introducing this Euclidean geometry course in
high schools was the geometry book by Charles Davies (1798–1876), a professor
at Columbia College, New York. Yet he attributes his work to that of the French
mathematician Adrien–Marie Legendre (1752–1833). However it got here, this
book provides a course in the deductive development of geometry, which has
gone through a number of changes over the past fifty years. But this book
captures these many modifications and is completely consistent with the current
mathematics standards in the United States.
The book has many features that will aid in truly appreciating the beauty of
geometry, with many interesting exercises—from rather simple to quite
challenging, separated by sectional rankings A, B, and C. It is particularly
noteworthy to consider how each theorem is presented and developed, often-
time through class exercises intended to explore the ensuing theorem. It would
be wise to consider these details as it will make the material far more intelligible.
Enrichment is a very important aspect of teaching any subject matter.
Towards that end we offer “Mathematical Excursions” throughout the book,
which are intended to allow the reader to expand in areas beyond the curriculum.
There are a number of other special features in the book such as historical notes
(“A Look at the Past”) that will help students appreciate the mathematics being
presented as well as some other shorter enriching topics under the heading of
“Something to Think About.”
With Chapter Thirteen the book extends beyond plane surfaces dealt with
previously to consider the surface areas and volumes of solids. The last two
chapters offer alternative approaches to geometry, one through the study of
vectors, and the other departing from Euclidean geometry into what is known as
non-Euclidean geometry.
We hope that you will enjoy this rather comprehensive treatment of
We hope that you will enjoy this rather comprehensive treatment of
geometry, which is the equivalent of the high school geometry course with some
features allowing the reader to expand beyond that realm.
Alfred S. Posamentier, April 2014
PREFACE / To the Student
This book was written for you to read and enjoy. It is a logical development of a
body of mathematical knowledge known as geometry. Included in this book are
topics from plane geometry, solid geometry, coordinate geometry, vector
geometry, and non-Euclidean geometry, the latitude of which can enhance your
interest of geometry.
How does one read and study a geometry book? You will find that each new
concept in the book is carefully developed and then clarified and reinforced with
many examples. In addition, there are over 2,000 illustrations to help you
visualize the conditions of the problems. Prior to each set of exercises, there are
numerous examples with their solutions completely worked out in detail. Before
attempting to start any assignment, you should read the text and study the sample
solutions that precede the exercises. The Class Exercises that appear in each
section will also help you to understand the concepts of the sections more
clearly. Study this material with pencil at hand, supplying illustrations where
they are not provided. If necessary, reread the section that precedes the
exercises.
Within each chapter are inserts to be used as supplementary reading, to
enrich your understanding of geometry. These come under the headings A Look
at the Past and Something to Think About. You will find the course far more
interesting if you take the time to read and explore these ideas. Further
supplementary material is given at the conclusion of selected chapters in the
sections entitled Mathematical Excursion. This material is definitely more
challenging and has been prepared for those of you with special interests and
abilities in mathematics.
Also included at the end of each chapter is a vocabulary list, a set of review
exercises, a chapter test, and topics for suggested research. These should help
you to organize and evaluate your knowledge before going on to new work.
The cumulative review at the close of every three chapters is a unique feature
of this text. The reviews are designed for self-evaluation, and you should enjoy
the experience of working through these.
The study of mathematics can be an exciting and rewarding experience. We
wish you luck as you begin this study of geometry.
The authors
PREFACE / To the Teacher
As indicated in the preface to the students, Geometry: Its Elements and Structure
has been prepared to present a logical development of a course in geometry that
emphasizes basic concepts and understandings of many branches of the tree of
geometry. It has been written with the reports and the recommendations of
various commissions and curriculum groups in mind, and is consistent with the
Common Core State Standards.
Both of the authors have a strong interest in secondary education and are
either currently teaching, or have taught, at this level, and are now involved in
the training of secondary school teachers of mathematics. Both are or were
members of the National Council of Teachers of Mathematics. Consequently, the
material presented is felt to be realistic in terms of what high school students can
be expected to absorb and learn. Furthermore there is an abundance of exercises,
reviews, and tests for constant and ongoing reinforcement and evaluation. The
considerable number of exercises allows you to spiral your homework
assignments for optimal individualization.
In addition, class exercises are provided in those places where they will be
helpful to the development and/or reinforcement of the lesson. Summaries are
placed within the chapters where they will be of greatest use to the individual
student.
Both the content and the development is based on the suggestions offered by
high school teachers and professors of mathematics. To provide for individual
differences, all sets of exercises have been graded, with most having levels A, B,
and C designated. This gradation of exercises, together with optional sections in
the book, allows it to be used for classes of varying abilities. A Solutions Manual
is also available from the publisher upon request.
End-of-chapter material is extensive and can be used in various ways, as
indicated by the annotations on these pages. The Mathematical Excursions at the
end of selected chapters are yet another means of providing enrichment for
gifted students, whereas the review exercises should prove helpful to slower
students.
We welcome all comments, criticisms, and suggestions from teachers and
students alike. We hope that you will enjoy teaching from this book as much as
we have enjoyed preparing it for you.
Alfred S. Posamentier, Robert L. Bannister
ABOUT THE AUTHORS
ALFRED S. POSAMENTIER is currently Dean of the School of Education and
Professor of Mathematics Education at Mercy College, New York, and was
previously Distinguished Lecturer at New York City College of Technology of
the City University of New York. He is Professor Emeritus of Mathematics
Education at The City College of The City University of New York, and former
Dean of the School of Education, where he was for forty years. He has been
Visiting Professor at several European universities in Austria, England,
Germany, the Czech Republic, and Poland, while at the University of Vienna he
was Fulbright Professor (1990). He has taken on numerous important leadership
positions in mathematics education locally. He was a member of the New York
State Education Commissioner’s Blue Ribbon Panel on the Math-A Regents
Exams, and the New York State Commissioner’s Mathematics Standards
Committee, which redefined the Standards for New York State, and he also
served on the New York City’s Schools’ Chancellor’s Math Advisory Panel. Dr.
Posamentier is the author and co-author of more than fifty-five mathematical
books for teachers, secondary and elementary school students, and the general
readership. He is also a frequent commentator in newspapers and journals on
topics relating to education.
ROBERT L. BANNISTER is the educational consultant to the President at St.
Louis University High School, St. Louis, Missouri. Prior to this, he served as
Mathematics teacher and then Associate Principal at Horton Watkins High
School, Ladue, Missouri before becoming the Principal at St. Louis University
High School. He has taught Middle School, High School and University
Mathematics classes and worked as a High School Mathematics editor. He
assisted in the development and formation of several Middle Schools in St.
Louis for under-served students. He also has served on a variety of Civic and
Cultural association’s Boards. He helped to develop an individualized and
innovative Mathematics program while at Horton Watkins High School as well
as contributed to and taught a Mathematical Research program for St. Louis
University’s graduate students. He is still involved in Mathematics classes at St.
Louis University High School when their teachers need assistance.
TABLE OF CONTENTS
Chapter 1
Introduction to Geometry
1-1 Review of Basic Terms
1-2 Arriving at Conclusions
1-3 Assumptions and Definitions
1-4 Lines
1-5 Angles
1-6 Polygons
1-7 Lines and Planes in Space
Chapter 2
Proof in Geometry
2-1 Basic Postulates
2-2 Plane, Space, and Angle Postulates
2-3 Using Postulates in Proof
2-4 Some Simple Proofs
2-5 Geometric Theorems
2-6 Two-Column Proofs
Chapter 3
Congruent Triangles
3-1 Congruence: An Equivalence Relation
3-2 Congruent Triangles
3-3 Congruence Postulates for Triangles
3-4 Isosceles and Equilateral Triangles
3-5 Overlapping Triangles
Chapter 4
Perpendicularity
4-1 Further Implications
4-2 Indirect Proof
4-3 Proving Two Pairs of Triangles Congruent
4-4 Perpendiculars in a Plane