Hemmerling E. M. - Fundamentals of College Geometry (2nd ed.).pdf - Geometry - Kuya

Fundamentals of

COLLEGE

GEOMET

~ I

SECOND

EDITION

II.

Edwin M. Hemmerling

Department

of Mathematics

Bakersfield

College

JOHN WILEY & SONS, New York 8 Chichisterl8

Brisbane 8 Toronto

Preface

Before revlsmg Fundamentals

of College Geometry, extensive

questionnaires

\rcre sent to users of the earlier edition.

A conscious effort has been made

in this edition to incorporate

the many fine suggestions

given the respondents

to the questionnaire.

At the same time, I have attempted

to preserve the

features that made the earlier edition so popular.

The postulational structure of the text has been strengthened. Some

definitions have been improved, making possible greater rigor in the develop-

ment of the theorems. Particular stress has been continued in observing the

distinction between equality and congruence. Symbols used for segments,

intervals, rays, and half-lines have been changed in order that the symbols

for the more common segment and ray will be easier to write.

However, a

symbol for the interval and half-line is introduced,

which will still logically

show their, relations to the segment and ray,

Fundamental

space concepts are introduced

throughout

the text in order

to preserve

continuity.

However,

the postulates

and theorems

on space

geometry are kept to a minimum

until Chapter

14. In this chapter, partic-

ular attention is given to mensuration

problems dealing with geometric solids.

Greater emphasis

has been placed on utilizing the principles

of deductive

logic covered

in Chapter

2 in deriving

geometric

truths

in subsequent

chapters.

Venn diagrams and truth tables have been expanded

at a number

of points throughout the text.

there is a wide vanance throughout

the Ul1lted States in the time spent in

geometry classes,

Approximately

two fifths of the classes meet three days a

week.

Another

two fifths meet five days each week,

The student

who

studies the first nine chapters of this text will have completed

a well-rounded

1970, by John Wiley & Sons, Ine.

minimum

course, including

all of the fundamental

concepts

of plane and

space geometry.

Each subsequent chapter in the book is written as a complete package,

none of which is essential to the study of any of the other last five chapters,

vet each will broaden

Reproduction or translation of any part of this work beyond that

permitted by Sections 107 or 108 of the 1976 United States Copy-

right Act without the permission of the copyright owner is unlaw-

ful. Requests for permission

the total background

of the student.

This will permit

or further information

should be

the instructor

considerable

latitude in adjusting

his course to the time avail-

Wiley & Sons, Inc.

addressed to the Permissions Department, J.ohn

able and to the needs of his students.

Each chapter contains several sets of summary tests.

20 19 18 17 16 15 14 13

Library of Congress Catalogue Card Number: 75-82969

SBN 47] 37034 7

Printed in the United States of America

These vary in type to

include true-false

tests, completion

tests, problems tests, and proofs tests.

key for these tests and the problem sets throughout

the text is available.

Januarv 1969

EdwinM. Hemmerling

Preface to First Edition

During the past decade the entire approach

to the teaching of geometry has

bccn undergoing

serious study by various nationally recognized

professional

groups. This book reflects many of their recommendations.

The style and objectives of this book are the same as those of my College

Plane Geometry, out of which it has grown.

Because I have added a signifi-

cant amount

of new material,

however, and have increased

the rigor em-

ployed, it has seemed desirable

to give the book a new title.

In Funda-

mentals of College Geometry, the presentation

of the su~ject has been strength-

encd by the early introduction

and continued

use of the language

and

symbolism of sets as a unifying concept.

This book is designed for a semester's work.

The student is introduced

the basic structure of geometry and is prepared to relate it to everyday

experience as well as to subsequent study of mathematics.

The value of the precise use of language

in stating definitions

and hypo-

theses and in developing

proofs is demonstrated.

The student is helped

to acquire an understanding

of deductive

thinking and a skill in applying it

to mathematical

situations.

He is also given experience

in the use of induc-

tion, analogy, and indirect methods of reasoning.

Abstract materials of geometry are related to experiences of daily life of

the student. He learns to search for undefined terms and axioms in such

areas of thinking as politics, sociology, and advertising.

Examples of circular

reasoning are studied.

In addition to providing for the promotion of proper attitudes, under-

standings, and appreciations, the book aids the student in learning to be

critical in his listening, reading, and thinking. He is taught not to accept

statements blindly but to think clearly before forming conclusions.

The chapter

on coordinate

geometry

relates

geometry

and algebra.

Properties

of geometric

figures are then determined

analytically with the aid

of algebra and the concept of one-to-one

correspondence.

A short chapter

on trigonometry

is given to relate ratio, similar polygons,

and coordinate

geometry.

Illustrative examples which aid in solving subsequent exercises are used

liberally throughout the book. The student is able to learn a great deal of

t he material without the assistance of an instructor.

Throughout

the book he

is afforded frequent

opportunities

for original and creative thinking.

Many

of the generous

supply of exercises include developments

which prepare

for theorems that appear later in the text.

The student is led to discover for

himself proofs that follow.

VII

Contents

The summary tests placed at the end of the book include completion,

true-

false, multiple-choice

items, and problems.

They afford the student and the

instructor a ready means of measuring

progress in the course.

Bakersfield, California, 1964

Edwin M. Hemmerling

I. Basic Elements of Geometry

2. Elementary

Logi.c

3. Deductive Reasoning

101

4. Congruence

- Congruent

Triangles

5. Parallel and Perpendicular

Lines

139

183

6. Polygons - Parallelograms

7. Circles

206

8. Proportion - Similar Polygons

9. Inequalities

245

283

10. Geometric Constructions

303

II. Geometric Loci

319

12. Areas of Polygons

340

13. Coordinate

Geometry

360

14. Areas and Volumes of Solids

388

------------

------.-

-----------------

Appendix

Greek Alphabet

417

419

Symbols and Abbreviations

419

Table 1. Square Roots

421

Properties of Real Number System

422

List of Postulates

423

Lists of Theorems and Corollaries

425

Answers to Exercises

437

Index

459

Vlll

111

Basic Elements of Geometry

1.1. Historical background of geometry. Geometry is a study of the pro-

perties and measurements of figures composed of points and lines. It is a

very old science and grew out of the needs of the people. The word geo-

metry is derived from the Greek words geo, meaning "earth," and metrein,

meaning "to measure." The early Egyptians and Babylonians (4000-3000

E.C.) were able to develop a collection of practical rules for measuring

simple

geometric figures and for determining

their properties.

These rules were obtained

inductively

over a period of centuries

of trial

<111( error.

They were not supported

by any evidence

of logical proof.

Applications

of these principles

were found in the building of the Pyramids

and the great Sphinx.

The irrigation systems devised by the early Egyptians indicate that they had

an adequate

knowledge

of geometry as it may be applied in land surveying.

The Babylonians

were using geometric figures in tiles, walls, and decorations

of their temples.

From Egypt and Babylonia the knowledge of geometry was taken to

Greece. From the Greek people we have gained some of the greatest con-

tributions to the advancement of mathematics. The Greek philosophers

studied geometry not only for utilitarian benefits derived but for the esthetic

and cultural advantages gained. The early Greeks thrived on a prosperous

sea trade. This sea trade brought them not only wealth but also knowledge

from other lands. These wealthy citizens of Greece had considerable time

for fashionable debates and study on various topics of cultural interest be-

cause they had slaves to do most of their routine work. Usually theories and

concepts brought back by returning seafarers from foreign lands made topics

for lengthy and spirited debate by the Greeks.

Hemmerling E. M. - Fundamentals of College Geometry (2nd ed.).pdf

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