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GIFT OF

ROBBINS'S NEW

PLANE GEOMETRY

BY

EDWARD RUTLEDGE ROBBINS, A.B.

FORMERLY OF LAWRENCEVILLE SCHOOL

AMERICAN BOOK COMPANY

NEW YORK CINCINNATI CHICAGO

COPYRIGHT, 1915, BY

EDWARD RUTLEDGE ROBBINS.

ROBBINS'S NEW PLANK GEOMETRY.

W. P. I

FOR THOSE WHOSE PRIVILEGE

IT MAY BE TO ACQUIRE A KNOWLEDGE OF

GEOMETRY

THIS VOLUME HAS BEEN WRITTEN

AND TO THE BOYS AND GIRLS WHO LEARN THE ANCIENT SCIENCE

FROM THESE PAGES, AND WHO ESTEEM THE POWER

OF CORRECT REASONING THE MORE

BECAUSE OF THE LOGIC OF

PURE GEOMETRY

THIS VOLUME IS DEDICATED

459819

PREFACE

THIS New Plane Geometry is not only the outgrowth, of the

author's long experience in teaching geometry, but has profited

further by suggestions from -teachers who have used Robbins's

" Plane Geometry " and by many of the recommendations of the

"National Committee of Fifteen." While many new and valu-

able features have been added in the reconstruction, yet all the

characteristics that met with widespread favor in the old book

have been retained.

Among the features of the book that make it sound and teach-

able may be mentioned the following :

1. The book has been written for the pupil. The objects sought

in the study of Geometry are (1) to train the mind to accept

only those statements as truth for which convincing reasons can

be provided, and (2) to cultivate a foresight that will appreci-

ate both the purpose in making a statement and the process of

reasoning by which the ultimate truth is established. Thus, the

study of this formal science should develop in the pupil the

ability to pursue argument coherently, and to establish geometric

truths in logical order. To meet the requirements of the various

degrees of intellectual capacity and maturity in every class, the

reason for every statement is not printed in full but is indicated

by a reference. The pupil who knows the reason need not con-

sult the paragraph cited; while the pupil who does not know it

may learn it by the reference. It is obvious that the greater

progress an individual makes in assimilating the subject and in

entering into its spirit, the less need there will be for the printed

reference.

2. Every effort has been made to stimulate the mental activity

of the pupil. To compel a young student, however, to supply his

vi PREFACE

own demonstrations frequently proves unprofitable as well as

arduous, and engenders in the learner a distaste for a study in

which he might otherwise take delight. This text does not aim

to produce accomplished geometricians at the completion of the

first book, but to aid the learner in his progress throughout the

volume, wherever experience has shown that he is likely to

require assistance. It is designed, under good instruction, to

develop a clear conception of the geometric idea, and to produce

at the end of the course a rational individual and a friend of this

particular science.

3. The theorems and their demonstrations the real subject-

matter of Geometry are introduced as early in the study as

possible.

4. The simple fundamental truths are explained instead of

being formally demonstrated.

5. The original exercises are distinguished by their abundance,

their practical bearings upon the affairs of life, their careful

gradation and classification, and their independence. Every ex-

ercise can be solved or demonstrated without the use of any other

exercise. Only the truths in the numbered paragraphs are nec-

essary in working originals.

6. The exercises are introduced as near as practicable to the

theorems to which they apply.

7. Emphasis is given to the discussion of original constructions.

8. The summaries will be found a valuable aid in reviews.

9. The historical notes give the pupil a knowledge of the devel-

opment of the science of geometry and add interest to the study.

10. The attractive open page will appeal alike to pupils and

to teachers.

The author sincerely desires to extend his thanks to those

friends and fellow teachers who, by suggestion and encourage-

ment, have inspired him in the preparation of these pages.

EDWARD K. BOBBINS.

CONTENTS

INTRODUCTION

PAGE

DEFINITIONS 1

ANGLES 2

TRIANGLES 4

CONGRUENCE ...... 5

SYMBOLS ............ 6

AXIOMS 6

POSTULATES . 7

EXERCISES 9

BOOK I. ANGLES, LINES, RECTILINEAR FIGURES

PRELIMINARY THEOREMS . . . 13

THEOREMS AND DEMONSTRATIONS 15

TRIANGLES 15

PARALLEL LINES . .20

QUADRILATERALS 47

POLYGONS 60

SYMMETRY 65

CONCERNING ORIGINAL EXERCISES . . . .... 68

SUMMARY. GENERAL DIRECTIONS FOR ATTACKING ORIGINALS . 68

ORIGINAL EXERCISES 70

BOOK II. THE CIRCLE

DEFINITIONS 75

PRELIMINARY THEOREMS 77

THEOREMS AND DEMONSTRATIONS 78

SUMMARY 94

ORIGINAL EXERCISES . 95

vii

vi PREFACE

own demonstrations frequently proves unprofitable as well as

arduous, and engenders in the learner a distaste for a study in

which he might otherwise take delight. This text does not aim

to produce accomplished geometricians at the completion of the

first book, but to aid the learner in his progress throughout the

volume, wherever experience has shown that he is likely to

require assistance. It is designed, under good instruction, to

develop a clear conception of the geometric idea, and to produce

at the end of the course a rational individual and a friend of this

particular science.

3. The theorems and their demonstrations the real subject-

matter of Geometry are introduced as early in the study as

possible.

4. The simple fundamental truths are explained instead of

being formally demonstrated.

5. The original exercises are distinguished by their abundance,

their practical bearings upon the affairs of life, their careful

gradation and classification, and their independence. Every ex-

ercise can be solved or demonstrated without the use of any other

exercise. Only the truths in the numbered paragraphs are nec-

essary in working originals.

6. The exercises are introduced as near as practicable to the

theorems to which they apply.

7. Emphasis is given to the discussion of original constructions.

8. The summaries will be found a valuable aid in reviews.

9. The historical notes give the pupil a knowledge of the devel-

opment of the science of geometry and add interest to the study.

10. The attractive open page will appeal alike to pupils and

to teachers.

The author sincerely desires to extend his thanks to those

friends and fellow teachers who, by suggestion and encourage-

ment, have inspired him in the preparation of these pages.

EDWARD R. ROBBINS.

CONTENTS

INTRODUCTION

PAGE

DEFINITIONS 1

ANGLES 2

TRIANGLES 4

CONGRUENCE ...... 5

SYMBOLS . 6

AXIOMS . 6

POSTULATES 7

EXERCISES 9

BOOK I. ANGLES, LINES, RECTILINEAR FIGURES

PRELIMINARY THEOREMS 13

THEOREMS AND DEMONSTRATIONS 15

TRIANGLES 15

PARALLEL LINES . .20

QUADRILATERALS .......... 47

POLYGONS 60

SYMMETRY 65

CONCERNING ORIGINAL EXERCISES . . . .... 68

SUMMARY. GENERAL DIRECTIONS FOR ATTACKING ORIGINALS . 68

ORIGINAL EXERCISES 70

BOOK II. THE CIRCLE

DEFINITIONS 75

PRELIMINARY THEOREMS 77

THEOREMS AND DEMONSTRATIONS 78

SUMMARY 94

ORIGINAL EXERCISES . 95

vii

viii CONTENTS

PAGE

KINDS OF QUANTITIES. MEASUREMENT 96

ORIGINAL EXERCISES 109

Loci 114

ORIGINAL EXERCISES ON Loci 115

CONSTRUCTION PROBLEMS 117

ANALYSIS 131

ORIGINAL CONSTRUCTION PROBLEMS 132

BOOK III. PROPORTION. SIMILAR FIGURES

DEFINITIONS 143

THEOREMS AND DEMONSTRATIONS . 144

CONCERNING ORIGINALS . 175

ORIGINAL EXERCISES (NUMERICAL) 176

SUMMARY 180

ORIGINAL EXERCISES (THEOREMS) 180

CONSTRUCTION PROBLEMS 186

ORIGINAL CONSTRUCTION PROBLEMS . . . . . . 190

BOOK IV. AREAS

THEOREMS AND DEMONSTRATIONS 193

FORMULAS 207

ORIGINAL EXERCISES (NUMERICAL) 210

CONSTRUCTION PROBLEMS 213

ORIGINAL CONSTRUCTION PROBLEMS 221

BOOK V. REGULAR POLYGONS. CIRCLES

THEOREMS AND DEMONSTRATIONS 225

ORIGINAL EXERCISES (THEOREMS) 239

CONSTRUCTION PROBLEMS 242

FORMULAS 245

ORIGINAL EXERCISES (NUMERICAL) 248

ORIGINAL CONSTRUCTION PROBLEMS 253

MAXIMA AND MINIMA 254

ORIGINAL EXERCISES 259

INDEX . . .261

PLANE GEOMETRY

INTRODUCTION

1. Geometry is a science which treats of the measure-

ment of magnitudes.

2. A point is that which has position but not magnitude.

3. A line is that which has length but no other magni-

tude.

4. A straight line is a line which is determined (fixed in

position) by any two of its points. That is, two lines that

coincide entirely, if they coincide at any two points, are

straight lines.

5. A rectilinear figure is a figure containing straight lines

and no others.

6. A surface is that which has length and breadth but no

other magnitude.

7. A plane is a surface in which if any two points are

taken, the straight line connecting them lies wholly in that

surface.

8. Plane Geometry is a science which treats of the proper-

ties of magnitudes in a plane.

9. A solid is that which has length, breadth, and thick-

ness. A solid is that which occupies space.

10. Boundaries. The boundaries (or boundary) of a solid

are surfaces. The boundaries (or boundary) of a surface

1

PLANE GEOMETRY

;aJr,e' 'filler : -'T-he boundaries of a line are points. These

boundaries can be no part of the things they limit. A sur-

face is no part of a solid ; a line is no part of a surface ; a

point is no part of a line.

11. Motion. If a point moves, its path is a line. Hence,

if a point moves, it generates (describes or traces) a line ; if

a line moves (except upon itself), it generates a surface ;

if a surface moves (except upon itself), it generates a solid.

NOTE. Unless otherwise specified the word " line " means straight line.

ANGLES

ANGLE ADJACENT VERTICAL ANGLES RIGHT ANGLES

ANGLES PERPENDICULAR

12. A plane angle is the amount of divergence of two

straight lines that meet. The lines are called the sides

of the angle. The vertex of an angle is the point at which

the lines meet.

13. Adjacent angles are two angles that have the same

vertex and a common side between them.

14. Vertical angles are two angles- that have the same

vertex, the sides of one being prolongations of the sides of

the other.

16. If one straight line meets another and makes the ad-

jacent angles equal, the angles are right angles.

16. One line is perpendicular to another if they meet at

right angles. Either line is perpendicular to the other. The

point at which the lines meet is the foot of the perpendicular.

Oblique lines are lines that meet but are not perpendicular.

INTRODUCTION . 3

17. A straight angle is an angle whose sides lie in the

same straight line, but extend in opposite directions from

the vertex.

OBTUSE ACUTE COMPLEMENTARY SUPPLEMENTARY

ANGLE ANGLE ANGLES ANGLES

18. An obtuse angle is an angle that is greater than a

right angle. An acute angle is an angle that is less than

a right angle. An oblique angle is any angle that is not a

right angle.

19. Two angles are complementary if their sum is equal to

one right angle. Two angles are supplementary if their sum

is equal to two right angles. Thus, the complement of an

angle is the difference between one right angle and the given

angle. The supplement of an angle is the difference between

.two right angles and the given angle.

20. A degree is one ninetieth of a right angle. The

degree is the familiar unit used in measuring angles. It is

evident that there are 90 in a right angle ; 180 in two

right angles, or a straight angle ; 360 in four right angles.

There are 60 minutes (60') in one degree, and 60 seconds (60") in one

minute.

21. Parallel lines are straight lines that lie in the same

plane and that never meet, however far they are extended in

either direction.

22. Notation. A point is usually denoted by a capital letter, placed

near it. A line is denoted by two capital letters, placed one at each end,

or one at each of two of its points. Its length is sometimes represented

advantageously by a small letter written near it. Thus, the line AB ;

the line RS\ the line m.

R S m

4 PLANE GEOMETRY

There are various ways of naming angles. Sometimes three capital

letters are used, one on each side of the angle and one at the vertex ;

sometimes a small letter or a figure is placed within the angle. The

symbol for angle is Z.

M

ZAMXo*

/.XMA on AM

X O C

Z.a AND Z .BOO,

NOT ZO

In naming an angle by the use of three letters, the vertex letter is al-

ways placed between the others. Thus the A above are Z.AMX or

Z KM A, Z a, Z BOC, Z.x,/.APR,Z. APS, Z BPR, Z TPB, Z 5, etc.

In the above figure Z x = Z 5. The size of an angle depends on the

amount of divergence between its sides, and not upon their length.

An angle is said to be included by its sides. An angle is

bisected by a line drawn through the vertex and dividing

the angle into two equal angles.

TRIANGLES

23. A triangle is a portion of a plane bounded by three

straight lines. These lines are the sides. The vertices of a

triangle are the three points at which the sides intersect.

The angles of a triangle are the three angles at the three

vertices. Each side of a triangle has two angles adjoining

it. The symbol for triangle is A.

ISOSCELES A EQUILATERAL A BIGHT A OBTUSE A ACUTE A

EQUIANGULAR A SCALENE &

INTRODUCTION 5

The base of a triangle is the side on which the figure ap-

pears to stand. The vertex of a triangle is the vertex op-

posite the base. The vertex angle is the angle opposite the

base.

24. Kinds of triangles :

A scalene triangle is a triangle no two sides of which are equal.

An isosceles triangle is a triangle two sides of which are equal.

An equilateral triangle is a triangle all sides of which are equal.

A right triangle is a triangle one angle of which is a right angle.

An obtuse triangle is a triangle one angle of which is an obtuse angle,

An acute triangle is a triangle all angles of which are acute angles.

An equiangular triangle is a triangle all angles of which are equal.

25. The hypotenuse of a right triangle is the side opposite

the right angle. The sides forming the right angle are called

legs.

CONGRUENCE

26. Two geometric figures are said to be equal if they

have the same size or magnitude.

Two geometric figures are said to be congruent if, when

one is superposed upon the other, they coincide in all respects.

The corresponding parts of congruent figures are equal,

and are called homologous parts.

27. Homologous parts of congruent figures are equal.

If the triangles DEF and HIJ are congruent,

Z.D\$> homologous to and = to Z. H ;

DE is homologous to and = to HI;

Z E is homologous to and = to ^ /;

EF is homologous to and = to IJ.

NOTE. Congruent figures have the

same shape as well as the same size,

whereas equal figures do not necessarily have the same shape.

Ex.1. What is the complement of an angle of 35? 48? 80?

75 50' ? 8 20' ?

Ex. 2. What is the supplement of an angle of 100? 50? 148?

121 30'? 10 40'?

6

PLANE GEOMETRY

28. A curve or curved line, is a line no part of which is

straight.

A circle is a plane curve all points of which are equally

distant from a point in the plane, called the center.

An arc is any part of a circle.

A radius is a straight line from the center to any point of

the circle.

A diameter is a straight line containing the center and

having its extremities in the circle.

The length of the circle is called the circumference.

29. Symbols. The usual symbols and abbreviations em-

ployed in geometry are the following :

+ plus.

minus.

= equals, is equal to,

equal.

= does not equal.

^ congruent, or is con-

gruent to.

> is greater than.

< is less than.

.'. hence, therefore,

consequently.

JL perpendicular.

Js perpendiculars.

AXIOM, POSTULATE, AND THEOREM

30. An axiom is a statement admitted without proof to be

true. It is a truth, received and assented to immediately.

31. AXIOMS.

1. Magnitudes that are equal to the same thing, or to equals,

are equal to each other.

2. If equals are added to, or subtracted from, equals, the results

are equal.

3. If equals are multiplied by, or divided by, equals, the results

are equal.

[Doubles of equals are equal; halves of equals are equal.]

O circle.

Ax.

axiom.

circles.

Hyp.

hypothesis.

Z angle.

comp.

complementary.

A angles.

supp.

supplementary.

rt. Z right angle.

Const.

construction.

rt. A right angles.

Cor.

corollary.

A triangle.

St.

straight.

& triangles.

rt.

right.

rt. & right triangles.

Def.

definition.

II parallel.

alt. .

alternate.

Us parallels.

int.

interior.

d parallelogram.

ext.

exterior.

17 parallelograms.

INTRODUCTION 7

4. The whole is equal to the sum of all of its parts.

5. The whole is greater than any of its parts.

6. A magnitude may be displaced by its equal in any process.

[Briefly called "substitution."]

7. If equals are added to, or subtracted from, unequals, the

results are unequal in the same order.

8. If unequals are added to unequals in the same order, the

results are unequal in that order.

9. If unequals are subtracted from equals, the results are un-

equal in the opposite order.

10. Doubles or halves of unequals are unequal in the same order.

Also, unequals multiplied by equals are unequal in the same

order.

11. If the first of three magnitudes is greater than the second,

and the second is greater than the third, the first is greater than

the third.

12. A straight line is the shortest line that can be drawn be-

tween two points.

13. Only one line can be drawn through a point parallel to a

given line.

14. A geometrical figure may be moved from one position to

another without any change in form or magnitude.

32. A postulate is something required to be done, the pos-

sibility of which is admitted without proof.

33. POSTULATES.

1. It is possible to draw a straight line from any point to any

other point.

2. It is possible to extend (prolong or produce) a straight line

indefinitely, or to terminate it at any point.

8 PLANE GEOMETRY

34. A geometric proof or demonstration is a logical course

of reasoning by which a truth becomes evident.

35. A theorem is a statement that requires proof.

In the case of the preliminary theorems which follow, the

proof is very simple ; but as these theorems are not admitted

without proof they cannot be classified with the axioms.

A corollary is a truth immediately evident, or readily es-

tablished from some other truth or truths.

A proposition, in geometry, is the statement of a theorem

to be proved or a problem to be solved.

Ex. 1. Draw an Z ABC. In /. ABC draw line BD. .

What does Z ABD + Z. DEC equal ?

What does Z ABC - ZABD equal?

Ex. 2. In a rt. Z.ABC draw line BD.

If ZABD = 25, how many degrees are there in Z DB C ?

How many degrees are there in the complement of an angle of 38 ?

How many degrees are there in the supplement ?

Ex. 3. Draw a straight line AB and take a point X on it.

What line does AX + BX equal?

What line does AB - BX equal?

Ex. 4. Draw a straight line AB and prolong it to X so ih&iBX = AB.

Prolong it so that AB = AX.

Historical Note. Probably as early as 3000 B.C. the Egyptians had

some knowledge of geometric truths. The construction of the great

pyramids required an acquaintance with the relations of geometry.

This knowledge, however vague it may have been, was, according to

Herodotus, employed in determining the amount of land washed away

by the river Nile, during the reign of Rameses II (1400 B.C.).

The Greeks, however, were the first to study geometry as a logical

science. They enunciated theorems and demonstrated them, they pro-

pounded problems and solved them as early as 300 B.C., and, in a crude

way, two or three centuries earlier. To them belongs the credit of estab-

lishing a logical system of geometry that has survived, practically un-

changed, for twenty centuries.

INTRODUCTION

B

EXERCISES EMPLOYING THE TWO INSTRUMENTS OF

GEOMETRY

Aside from pencil and paper, the only instruments

necessary for the construction of geometrical dia-

grams are the ruler and the compasses.

Ex. 1. It is required to draw an equilateral tri-

angle upon a given line as base.

Suppose AB is the given base.

Required to draw an equilateral A upon it.

Using A as a center and AB as a radius, draw an arc. Using B

as a center and AB as a radius, draw another arc cutting the first one

at C. Draw AC and BC. The &ABC is an equi-

lateral A, and AB is its base.

Ex. 2. It is required to draw a triangle having its

three sides each equal to a given line.

Suppose the three given lines are a,b,c.

Required to draw a A having for its sides lines

equal to a, &, c-, respectively.

Draw a line RS to a. Using R as a center and

b as a radius, draw an arc. Using S as a center

and c as a radius, draw another arc cutting the first

arc at T. Draw straight lines R T and ST. A RST

is the A whose three sides are equal to the lines

a, b, c, respectively.

Ex. 3. It is required to find the midpoint of a

given straight line.

Given the straight line AB.

Required to find its midpoint.

Using A and B as centers and a radius sufficiently / j \

long, draw two arcs, intersecting at P and Q. AI r^-j B

Draw the straight line PQ cutting AB at M. \ I /

Point M is the midpoint of A B. \ \ /

Ex. 4. It is required to draw a perpendicular to a '"Q x

line from a point within the line.

Given the line CD and point P in it. ..K

Required to construct a _L to CD, at P.

Using P as center and any radius, draw two arcs ^ / I \ p

cutting CD at E and F. Now using E and F as

centers and a radius greater than before, draw two arcs intersecting at K.

Draw KP. This line KP is _L to CD at P.

BOBBINS'S NEW PLANE GEOM. 2

10

PLANE GEOMETRY

P

L

Ex. 6. It is required to draw a perpendicular to a

line from a point without the line.

Given line AB and point P, without it.

Required to draw a _L to A B from P.

Using P as center and a sufficient radius, draw A-

an arc cutting AB at C and D. Now using C and

D as centers and a sufficient radius, draw two arcs intersecting at E.

Draw PE, meeting AB at R. PR is the required _L to AB from P.

Ex. 6. It is required to bisect a given angle.

Given the Z ABC.

Required to bisect it.

Using vertex B as a center and any radius, draw

arc DE cutting BC at D and BA at E.

Using D and E as centers-and a sufficient radius, draw arcs intersect-

ing at F. Draw straight line BF. BF bisects the Z ABC.

Ex. 7. It is required to cpnstruct, at a given point on a given line, an

angle equal to a given angle.

Given line DE, point D in it,

and Z B.

Required to construct an Z at D,

equal to Z 5.

Using .B as a center and with any two distances as radii, draw an arc

cutting AB at F and another cutting BC at G.

Using D as a center and the same radii as before, draw one arc, and

another arc cutting DE at /.

Draw the straight line FG. Using / as a center and FG as a radius,

draw an arc cutting a former arc at H. Draw the straight lines HJ and

DHK.

Now the Z KDE = Z B.

Ex. 8. By the use of ruler and compasses, draw the following figures :

G

Ex. 9. Does it make any difference in these exercises, which lines are

drawn first? In Ex. 7 and Ex. 8 explain the order of the lines drawn.

INTRODUCTION 11

Ex. 10. Using the compasses only, draw the following figures:

Ex. 11. Draw the following figures :

Ex. 12. Draw the first of each of these three pairs of figures.

Can you explain the construction of the second figure in each pair ?

12

PLANE GEOMETRY

In this figure, ABCD is a square. On

the sides are measured the equal dis-

tances AE and BF, and CG and DH ; (

then the lines AG, BH, CE, and DF, are

drawn intersecting at a, b, c, d. The

figure abed is also .a square. This figure

is the basis of an Arabic design used for

parquet floors, etc.

In this figure, which is the basis of a

mosaic floor design, the radii of all com-

plete circles equal one fourth of the side

of the square ABCD. The radii of the

semicircles GflJ, IKR, etc., equal one

eighth of the side of the square.

In this figure ABD is an equilateral arch ?

and CD is its altitude. The several cen-

ters used are A, of arc BD and arc CE ; 5,

of arc AD and CF ; (7, of arcs AE and BF.

This figure is the basis of a common

Gothic window design.

NOTE. The letters " Q.E.D." are often annexed at the end of a demon-

stration and stand for "quod erat demonstrandum," which means, " which

was to be proved."

BOOK I

ANGLES, LINES, RECTILINEAR FIGURES

ROBBINS'S NEW

PLANE GEOMETRY

BY

EDWARD RUTLEDGE ROBBINS, A.B.

FORMERLY OF LAWRENCEVILLE SCHOOL

AMERICAN BOOK COMPANY

NEW YORK CINCINNATI CHICAGO

COPYRIGHT, 1915, BY

EDWARD RUTLEDGE ROBBINS.

ROBBINS'S NEW PLANK GEOMETRY.

W. P. I

FOR THOSE WHOSE PRIVILEGE

IT MAY BE TO ACQUIRE A KNOWLEDGE OF

GEOMETRY

THIS VOLUME HAS BEEN WRITTEN

AND TO THE BOYS AND GIRLS WHO LEARN THE ANCIENT SCIENCE

FROM THESE PAGES, AND WHO ESTEEM THE POWER

OF CORRECT REASONING THE MORE

BECAUSE OF THE LOGIC OF

PURE GEOMETRY

THIS VOLUME IS DEDICATED

459819

PREFACE

THIS New Plane Geometry is not only the outgrowth, of the

author's long experience in teaching geometry, but has profited

further by suggestions from -teachers who have used Robbins's

" Plane Geometry " and by many of the recommendations of the

"National Committee of Fifteen." While many new and valu-

able features have been added in the reconstruction, yet all the

characteristics that met with widespread favor in the old book

have been retained.

Among the features of the book that make it sound and teach-

able may be mentioned the following :

1. The book has been written for the pupil. The objects sought

in the study of Geometry are (1) to train the mind to accept

only those statements as truth for which convincing reasons can

be provided, and (2) to cultivate a foresight that will appreci-

ate both the purpose in making a statement and the process of

reasoning by which the ultimate truth is established. Thus, the

study of this formal science should develop in the pupil the

ability to pursue argument coherently, and to establish geometric

truths in logical order. To meet the requirements of the various

degrees of intellectual capacity and maturity in every class, the

reason for every statement is not printed in full but is indicated

by a reference. The pupil who knows the reason need not con-

sult the paragraph cited; while the pupil who does not know it

may learn it by the reference. It is obvious that the greater

progress an individual makes in assimilating the subject and in

entering into its spirit, the less need there will be for the printed

reference.

2. Every effort has been made to stimulate the mental activity

of the pupil. To compel a young student, however, to supply his

vi PREFACE

own demonstrations frequently proves unprofitable as well as

arduous, and engenders in the learner a distaste for a study in

which he might otherwise take delight. This text does not aim

to produce accomplished geometricians at the completion of the

first book, but to aid the learner in his progress throughout the

volume, wherever experience has shown that he is likely to

require assistance. It is designed, under good instruction, to

develop a clear conception of the geometric idea, and to produce

at the end of the course a rational individual and a friend of this

particular science.

3. The theorems and their demonstrations the real subject-

matter of Geometry are introduced as early in the study as

possible.

4. The simple fundamental truths are explained instead of

being formally demonstrated.

5. The original exercises are distinguished by their abundance,

their practical bearings upon the affairs of life, their careful

gradation and classification, and their independence. Every ex-

ercise can be solved or demonstrated without the use of any other

exercise. Only the truths in the numbered paragraphs are nec-

essary in working originals.

6. The exercises are introduced as near as practicable to the

theorems to which they apply.

7. Emphasis is given to the discussion of original constructions.

8. The summaries will be found a valuable aid in reviews.

9. The historical notes give the pupil a knowledge of the devel-

opment of the science of geometry and add interest to the study.

10. The attractive open page will appeal alike to pupils and

to teachers.

The author sincerely desires to extend his thanks to those

friends and fellow teachers who, by suggestion and encourage-

ment, have inspired him in the preparation of these pages.

EDWARD K. BOBBINS.

CONTENTS

INTRODUCTION

PAGE

DEFINITIONS 1

ANGLES 2

TRIANGLES 4

CONGRUENCE ...... 5

SYMBOLS ............ 6

AXIOMS 6

POSTULATES . 7

EXERCISES 9

BOOK I. ANGLES, LINES, RECTILINEAR FIGURES

PRELIMINARY THEOREMS . . . 13

THEOREMS AND DEMONSTRATIONS 15

TRIANGLES 15

PARALLEL LINES . .20

QUADRILATERALS 47

POLYGONS 60

SYMMETRY 65

CONCERNING ORIGINAL EXERCISES . . . .... 68

SUMMARY. GENERAL DIRECTIONS FOR ATTACKING ORIGINALS . 68

ORIGINAL EXERCISES 70

BOOK II. THE CIRCLE

DEFINITIONS 75

PRELIMINARY THEOREMS 77

THEOREMS AND DEMONSTRATIONS 78

SUMMARY 94

ORIGINAL EXERCISES . 95

vii

vi PREFACE

own demonstrations frequently proves unprofitable as well as

arduous, and engenders in the learner a distaste for a study in

which he might otherwise take delight. This text does not aim

to produce accomplished geometricians at the completion of the

first book, but to aid the learner in his progress throughout the

volume, wherever experience has shown that he is likely to

require assistance. It is designed, under good instruction, to

develop a clear conception of the geometric idea, and to produce

at the end of the course a rational individual and a friend of this

particular science.

3. The theorems and their demonstrations the real subject-

matter of Geometry are introduced as early in the study as

possible.

4. The simple fundamental truths are explained instead of

being formally demonstrated.

5. The original exercises are distinguished by their abundance,

their practical bearings upon the affairs of life, their careful

gradation and classification, and their independence. Every ex-

ercise can be solved or demonstrated without the use of any other

exercise. Only the truths in the numbered paragraphs are nec-

essary in working originals.

6. The exercises are introduced as near as practicable to the

theorems to which they apply.

7. Emphasis is given to the discussion of original constructions.

8. The summaries will be found a valuable aid in reviews.

9. The historical notes give the pupil a knowledge of the devel-

opment of the science of geometry and add interest to the study.

10. The attractive open page will appeal alike to pupils and

to teachers.

The author sincerely desires to extend his thanks to those

friends and fellow teachers who, by suggestion and encourage-

ment, have inspired him in the preparation of these pages.

EDWARD R. ROBBINS.

CONTENTS

INTRODUCTION

PAGE

DEFINITIONS 1

ANGLES 2

TRIANGLES 4

CONGRUENCE ...... 5

SYMBOLS . 6

AXIOMS . 6

POSTULATES 7

EXERCISES 9

BOOK I. ANGLES, LINES, RECTILINEAR FIGURES

PRELIMINARY THEOREMS 13

THEOREMS AND DEMONSTRATIONS 15

TRIANGLES 15

PARALLEL LINES . .20

QUADRILATERALS .......... 47

POLYGONS 60

SYMMETRY 65

CONCERNING ORIGINAL EXERCISES . . . .... 68

SUMMARY. GENERAL DIRECTIONS FOR ATTACKING ORIGINALS . 68

ORIGINAL EXERCISES 70

BOOK II. THE CIRCLE

DEFINITIONS 75

PRELIMINARY THEOREMS 77

THEOREMS AND DEMONSTRATIONS 78

SUMMARY 94

ORIGINAL EXERCISES . 95

vii

viii CONTENTS

PAGE

KINDS OF QUANTITIES. MEASUREMENT 96

ORIGINAL EXERCISES 109

Loci 114

ORIGINAL EXERCISES ON Loci 115

CONSTRUCTION PROBLEMS 117

ANALYSIS 131

ORIGINAL CONSTRUCTION PROBLEMS 132

BOOK III. PROPORTION. SIMILAR FIGURES

DEFINITIONS 143

THEOREMS AND DEMONSTRATIONS . 144

CONCERNING ORIGINALS . 175

ORIGINAL EXERCISES (NUMERICAL) 176

SUMMARY 180

ORIGINAL EXERCISES (THEOREMS) 180

CONSTRUCTION PROBLEMS 186

ORIGINAL CONSTRUCTION PROBLEMS . . . . . . 190

BOOK IV. AREAS

THEOREMS AND DEMONSTRATIONS 193

FORMULAS 207

ORIGINAL EXERCISES (NUMERICAL) 210

CONSTRUCTION PROBLEMS 213

ORIGINAL CONSTRUCTION PROBLEMS 221

BOOK V. REGULAR POLYGONS. CIRCLES

THEOREMS AND DEMONSTRATIONS 225

ORIGINAL EXERCISES (THEOREMS) 239

CONSTRUCTION PROBLEMS 242

FORMULAS 245

ORIGINAL EXERCISES (NUMERICAL) 248

ORIGINAL CONSTRUCTION PROBLEMS 253

MAXIMA AND MINIMA 254

ORIGINAL EXERCISES 259

INDEX . . .261

PLANE GEOMETRY

INTRODUCTION

1. Geometry is a science which treats of the measure-

ment of magnitudes.

2. A point is that which has position but not magnitude.

3. A line is that which has length but no other magni-

tude.

4. A straight line is a line which is determined (fixed in

position) by any two of its points. That is, two lines that

coincide entirely, if they coincide at any two points, are

straight lines.

5. A rectilinear figure is a figure containing straight lines

and no others.

6. A surface is that which has length and breadth but no

other magnitude.

7. A plane is a surface in which if any two points are

taken, the straight line connecting them lies wholly in that

surface.

8. Plane Geometry is a science which treats of the proper-

ties of magnitudes in a plane.

9. A solid is that which has length, breadth, and thick-

ness. A solid is that which occupies space.

10. Boundaries. The boundaries (or boundary) of a solid

are surfaces. The boundaries (or boundary) of a surface

1

PLANE GEOMETRY

;aJr,e' 'filler : -'T-he boundaries of a line are points. These

boundaries can be no part of the things they limit. A sur-

face is no part of a solid ; a line is no part of a surface ; a

point is no part of a line.

11. Motion. If a point moves, its path is a line. Hence,

if a point moves, it generates (describes or traces) a line ; if

a line moves (except upon itself), it generates a surface ;

if a surface moves (except upon itself), it generates a solid.

NOTE. Unless otherwise specified the word " line " means straight line.

ANGLES

ANGLE ADJACENT VERTICAL ANGLES RIGHT ANGLES

ANGLES PERPENDICULAR

12. A plane angle is the amount of divergence of two

straight lines that meet. The lines are called the sides

of the angle. The vertex of an angle is the point at which

the lines meet.

13. Adjacent angles are two angles that have the same

vertex and a common side between them.

14. Vertical angles are two angles- that have the same

vertex, the sides of one being prolongations of the sides of

the other.

16. If one straight line meets another and makes the ad-

jacent angles equal, the angles are right angles.

16. One line is perpendicular to another if they meet at

right angles. Either line is perpendicular to the other. The

point at which the lines meet is the foot of the perpendicular.

Oblique lines are lines that meet but are not perpendicular.

INTRODUCTION . 3

17. A straight angle is an angle whose sides lie in the

same straight line, but extend in opposite directions from

the vertex.

OBTUSE ACUTE COMPLEMENTARY SUPPLEMENTARY

ANGLE ANGLE ANGLES ANGLES

18. An obtuse angle is an angle that is greater than a

right angle. An acute angle is an angle that is less than

a right angle. An oblique angle is any angle that is not a

right angle.

19. Two angles are complementary if their sum is equal to

one right angle. Two angles are supplementary if their sum

is equal to two right angles. Thus, the complement of an

angle is the difference between one right angle and the given

angle. The supplement of an angle is the difference between

.two right angles and the given angle.

20. A degree is one ninetieth of a right angle. The

degree is the familiar unit used in measuring angles. It is

evident that there are 90 in a right angle ; 180 in two

right angles, or a straight angle ; 360 in four right angles.

There are 60 minutes (60') in one degree, and 60 seconds (60") in one

minute.

21. Parallel lines are straight lines that lie in the same

plane and that never meet, however far they are extended in

either direction.

22. Notation. A point is usually denoted by a capital letter, placed

near it. A line is denoted by two capital letters, placed one at each end,

or one at each of two of its points. Its length is sometimes represented

advantageously by a small letter written near it. Thus, the line AB ;

the line RS\ the line m.

R S m

4 PLANE GEOMETRY

There are various ways of naming angles. Sometimes three capital

letters are used, one on each side of the angle and one at the vertex ;

sometimes a small letter or a figure is placed within the angle. The

symbol for angle is Z.

M

ZAMXo*

/.XMA on AM

X O C

Z.a AND Z .BOO,

NOT ZO

In naming an angle by the use of three letters, the vertex letter is al-

ways placed between the others. Thus the A above are Z.AMX or

Z KM A, Z a, Z BOC, Z.x,/.APR,Z. APS, Z BPR, Z TPB, Z 5, etc.

In the above figure Z x = Z 5. The size of an angle depends on the

amount of divergence between its sides, and not upon their length.

An angle is said to be included by its sides. An angle is

bisected by a line drawn through the vertex and dividing

the angle into two equal angles.

TRIANGLES

23. A triangle is a portion of a plane bounded by three

straight lines. These lines are the sides. The vertices of a

triangle are the three points at which the sides intersect.

The angles of a triangle are the three angles at the three

vertices. Each side of a triangle has two angles adjoining

it. The symbol for triangle is A.

ISOSCELES A EQUILATERAL A BIGHT A OBTUSE A ACUTE A

EQUIANGULAR A SCALENE &

INTRODUCTION 5

The base of a triangle is the side on which the figure ap-

pears to stand. The vertex of a triangle is the vertex op-

posite the base. The vertex angle is the angle opposite the

base.

24. Kinds of triangles :

A scalene triangle is a triangle no two sides of which are equal.

An isosceles triangle is a triangle two sides of which are equal.

An equilateral triangle is a triangle all sides of which are equal.

A right triangle is a triangle one angle of which is a right angle.

An obtuse triangle is a triangle one angle of which is an obtuse angle,

An acute triangle is a triangle all angles of which are acute angles.

An equiangular triangle is a triangle all angles of which are equal.

25. The hypotenuse of a right triangle is the side opposite

the right angle. The sides forming the right angle are called

legs.

CONGRUENCE

26. Two geometric figures are said to be equal if they

have the same size or magnitude.

Two geometric figures are said to be congruent if, when

one is superposed upon the other, they coincide in all respects.

The corresponding parts of congruent figures are equal,

and are called homologous parts.

27. Homologous parts of congruent figures are equal.

If the triangles DEF and HIJ are congruent,

Z.D\$> homologous to and = to Z. H ;

DE is homologous to and = to HI;

Z E is homologous to and = to ^ /;

EF is homologous to and = to IJ.

NOTE. Congruent figures have the

same shape as well as the same size,

whereas equal figures do not necessarily have the same shape.

Ex.1. What is the complement of an angle of 35? 48? 80?

75 50' ? 8 20' ?

Ex. 2. What is the supplement of an angle of 100? 50? 148?

121 30'? 10 40'?

6

PLANE GEOMETRY

28. A curve or curved line, is a line no part of which is

straight.

A circle is a plane curve all points of which are equally

distant from a point in the plane, called the center.

An arc is any part of a circle.

A radius is a straight line from the center to any point of

the circle.

A diameter is a straight line containing the center and

having its extremities in the circle.

The length of the circle is called the circumference.

29. Symbols. The usual symbols and abbreviations em-

ployed in geometry are the following :

+ plus.

minus.

= equals, is equal to,

equal.

= does not equal.

^ congruent, or is con-

gruent to.

> is greater than.

< is less than.

.'. hence, therefore,

consequently.

JL perpendicular.

Js perpendiculars.

AXIOM, POSTULATE, AND THEOREM

30. An axiom is a statement admitted without proof to be

true. It is a truth, received and assented to immediately.

31. AXIOMS.

1. Magnitudes that are equal to the same thing, or to equals,

are equal to each other.

2. If equals are added to, or subtracted from, equals, the results

are equal.

3. If equals are multiplied by, or divided by, equals, the results

are equal.

[Doubles of equals are equal; halves of equals are equal.]

O circle.

Ax.

axiom.

circles.

Hyp.

hypothesis.

Z angle.

comp.

complementary.

A angles.

supp.

supplementary.

rt. Z right angle.

Const.

construction.

rt. A right angles.

Cor.

corollary.

A triangle.

St.

straight.

& triangles.

rt.

right.

rt. & right triangles.

Def.

definition.

II parallel.

alt. .

alternate.

Us parallels.

int.

interior.

d parallelogram.

ext.

exterior.

17 parallelograms.

INTRODUCTION 7

4. The whole is equal to the sum of all of its parts.

5. The whole is greater than any of its parts.

6. A magnitude may be displaced by its equal in any process.

[Briefly called "substitution."]

7. If equals are added to, or subtracted from, unequals, the

results are unequal in the same order.

8. If unequals are added to unequals in the same order, the

results are unequal in that order.

9. If unequals are subtracted from equals, the results are un-

equal in the opposite order.

10. Doubles or halves of unequals are unequal in the same order.

Also, unequals multiplied by equals are unequal in the same

order.

11. If the first of three magnitudes is greater than the second,

and the second is greater than the third, the first is greater than

the third.

12. A straight line is the shortest line that can be drawn be-

tween two points.

13. Only one line can be drawn through a point parallel to a

given line.

14. A geometrical figure may be moved from one position to

another without any change in form or magnitude.

32. A postulate is something required to be done, the pos-

sibility of which is admitted without proof.

33. POSTULATES.

1. It is possible to draw a straight line from any point to any

other point.

2. It is possible to extend (prolong or produce) a straight line

indefinitely, or to terminate it at any point.

8 PLANE GEOMETRY

34. A geometric proof or demonstration is a logical course

of reasoning by which a truth becomes evident.

35. A theorem is a statement that requires proof.

In the case of the preliminary theorems which follow, the

proof is very simple ; but as these theorems are not admitted

without proof they cannot be classified with the axioms.

A corollary is a truth immediately evident, or readily es-

tablished from some other truth or truths.

A proposition, in geometry, is the statement of a theorem

to be proved or a problem to be solved.

Ex. 1. Draw an Z ABC. In /. ABC draw line BD. .

What does Z ABD + Z. DEC equal ?

What does Z ABC - ZABD equal?

Ex. 2. In a rt. Z.ABC draw line BD.

If ZABD = 25, how many degrees are there in Z DB C ?

How many degrees are there in the complement of an angle of 38 ?

How many degrees are there in the supplement ?

Ex. 3. Draw a straight line AB and take a point X on it.

What line does AX + BX equal?

What line does AB - BX equal?

Ex. 4. Draw a straight line AB and prolong it to X so ih&iBX = AB.

Prolong it so that AB = AX.

Historical Note. Probably as early as 3000 B.C. the Egyptians had

some knowledge of geometric truths. The construction of the great

pyramids required an acquaintance with the relations of geometry.

This knowledge, however vague it may have been, was, according to

Herodotus, employed in determining the amount of land washed away

by the river Nile, during the reign of Rameses II (1400 B.C.).

The Greeks, however, were the first to study geometry as a logical

science. They enunciated theorems and demonstrated them, they pro-

pounded problems and solved them as early as 300 B.C., and, in a crude

way, two or three centuries earlier. To them belongs the credit of estab-

lishing a logical system of geometry that has survived, practically un-

changed, for twenty centuries.

INTRODUCTION

B

EXERCISES EMPLOYING THE TWO INSTRUMENTS OF

GEOMETRY

Aside from pencil and paper, the only instruments

necessary for the construction of geometrical dia-

grams are the ruler and the compasses.

Ex. 1. It is required to draw an equilateral tri-

angle upon a given line as base.

Suppose AB is the given base.

Required to draw an equilateral A upon it.

Using A as a center and AB as a radius, draw an arc. Using B

as a center and AB as a radius, draw another arc cutting the first one

at C. Draw AC and BC. The &ABC is an equi-

lateral A, and AB is its base.

Ex. 2. It is required to draw a triangle having its

three sides each equal to a given line.

Suppose the three given lines are a,b,c.

Required to draw a A having for its sides lines

equal to a, &, c-, respectively.

Draw a line RS to a. Using R as a center and

b as a radius, draw an arc. Using S as a center

and c as a radius, draw another arc cutting the first

arc at T. Draw straight lines R T and ST. A RST

is the A whose three sides are equal to the lines

a, b, c, respectively.

Ex. 3. It is required to find the midpoint of a

given straight line.

Given the straight line AB.

Required to find its midpoint.

Using A and B as centers and a radius sufficiently / j \

long, draw two arcs, intersecting at P and Q. AI r^-j B

Draw the straight line PQ cutting AB at M. \ I /

Point M is the midpoint of A B. \ \ /

Ex. 4. It is required to draw a perpendicular to a '"Q x

line from a point within the line.

Given the line CD and point P in it. ..K

Required to construct a _L to CD, at P.

Using P as center and any radius, draw two arcs ^ / I \ p

cutting CD at E and F. Now using E and F as

centers and a radius greater than before, draw two arcs intersecting at K.

Draw KP. This line KP is _L to CD at P.

BOBBINS'S NEW PLANE GEOM. 2

10

PLANE GEOMETRY

P

L

Ex. 6. It is required to draw a perpendicular to a

line from a point without the line.

Given line AB and point P, without it.

Required to draw a _L to A B from P.

Using P as center and a sufficient radius, draw A-

an arc cutting AB at C and D. Now using C and

D as centers and a sufficient radius, draw two arcs intersecting at E.

Draw PE, meeting AB at R. PR is the required _L to AB from P.

Ex. 6. It is required to bisect a given angle.

Given the Z ABC.

Required to bisect it.

Using vertex B as a center and any radius, draw

arc DE cutting BC at D and BA at E.

Using D and E as centers-and a sufficient radius, draw arcs intersect-

ing at F. Draw straight line BF. BF bisects the Z ABC.

Ex. 7. It is required to cpnstruct, at a given point on a given line, an

angle equal to a given angle.

Given line DE, point D in it,

and Z B.

Required to construct an Z at D,

equal to Z 5.

Using .B as a center and with any two distances as radii, draw an arc

cutting AB at F and another cutting BC at G.

Using D as a center and the same radii as before, draw one arc, and

another arc cutting DE at /.

Draw the straight line FG. Using / as a center and FG as a radius,

draw an arc cutting a former arc at H. Draw the straight lines HJ and

DHK.

Now the Z KDE = Z B.

Ex. 8. By the use of ruler and compasses, draw the following figures :

G

Ex. 9. Does it make any difference in these exercises, which lines are

drawn first? In Ex. 7 and Ex. 8 explain the order of the lines drawn.

INTRODUCTION 11

Ex. 10. Using the compasses only, draw the following figures:

Ex. 11. Draw the following figures :

Ex. 12. Draw the first of each of these three pairs of figures.

Can you explain the construction of the second figure in each pair ?

12

PLANE GEOMETRY

In this figure, ABCD is a square. On

the sides are measured the equal dis-

tances AE and BF, and CG and DH ; (

then the lines AG, BH, CE, and DF, are

drawn intersecting at a, b, c, d. The

figure abed is also .a square. This figure

is the basis of an Arabic design used for

parquet floors, etc.

In this figure, which is the basis of a

mosaic floor design, the radii of all com-

plete circles equal one fourth of the side

of the square ABCD. The radii of the

semicircles GflJ, IKR, etc., equal one

eighth of the side of the square.

In this figure ABD is an equilateral arch ?

and CD is its altitude. The several cen-

ters used are A, of arc BD and arc CE ; 5,

of arc AD and CF ; (7, of arcs AE and BF.

This figure is the basis of a common

Gothic window design.

NOTE. The letters " Q.E.D." are often annexed at the end of a demon-

stration and stand for "quod erat demonstrandum," which means, " which

was to be proved."

BOOK I

ANGLES, LINES, RECTILINEAR FIGURES