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GEOMETRY MANUAL
VOLUME 2
Elementary II
Edited by
Dr. Pamela Zell Rigg
and Jane Wechsler
©Copyright 2004
Montessori Research and Development
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Geometry Manual Volume 2
TABLE OF CONTENTS
THE STUDY OF GEOMETRY: LEVEL II AN INTRODUCTION ........................................... 1
THE STUDY OF GEOMETRY: LEVEL II FLOW CHART ..................................................... 2
CHAPTER 1: CLASSIFIED NOMENCLATURE
BASIC CONCEPTS - POINT, LINE, SURFACE, SOLID ....................................................... 4
STUDY OF THE LINE ................................................................................................... 7
STUDY OF ANGLES ...................................................................................................10
MEASUREMENT OF ANGLES ......................................................................................12
FRACTION/DEGREE/TYPE OF ANGLE CHART ..............................................................17
MEASUREMENT OF ANGLES WITH A REGULAR PROTRACTOR ....................................18
THE FORMATION OF REGIONS - SIMPLE CLOSED CURVE FIGURES AND POLYGONS ...21
STUDY OF THE TRIANGLE .........................................................................................23
STUDY OF TRIANGLES ACCORDING TO SIDES ...........................................................25
STUDY OF TRIANGLES ACCORDING TO ANGLES .......................................................28
STUDY OF TRIANGLES ACCORDING TO ANGLES AND SIDES......................................30
THE RIGHT TRIANGLE ...............................................................................................32
THE EQUILATERAL TRIANGLE ....................................................................................34
THE HEIGHT OR ALTITUDE OF A TRIANGLE ...............................................................36
ORTHOCENTER OF A TRIANGLE ................................................................................39
CONSTRUCT TRIANGLES WITH A COMPASS ..............................................................41
STUDY OF QUADRILATERALS ACCORDING TO SIDES AND ANGLES ...........................43
THE HEIGHT OR ALTITUDE OF A QUADRILATERAL .....................................................48
POLYGONS - REGULAR AND IRREGULAR ................................................................. 52
STUDY OF QUADRILATERALS - EQUIANGULAR AND EQUILATERAL ...........................55
STUDY OF TRIANGLES - EQUIANGULAR AND EQUILATERAL ......................................56
THE DIAGONAL ........................................................................................................57
THE DIAGONAL ........................................................................................................63
THE TRANSVERSAL AND ITS ANGLES .........................................................................64
ALTERNATE ANGLES ARE EQUAL ................................................................................66
INTERIOR ANGLES ON THE SAME SIDE OF THE TRANSVERSAL .................................. 68
EXTERIOR ANGLES ON THE SAME SIDE OF THE TRANSVERSAL .................................. 69
ANGLES ON THE SAME SIDE OF THE TRANSVERSAL ...................................................70
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Geometry Manual Volume 2
CORRESPONDING ANGLES ..............................................................................72
ADJACENT ANGLES ..........................................................................................74
VERTICAL ANGLES ...........................................................................................77
TWO PARALLEL STRAIGHT LINES CUT BY A TRANSVERSAL ................................81
ERATOSTHENES’ GREAT DISCOVERY: THE CIRCUMFERENCE OF THE EARTH .......83
THE CIRCLE:
NOMENCLATURE AND PROPERTIES ............................................................ 85
RELATIONSHIP OF A STRAIGHT LINE AND CIRCUMFERENCE - EXTERNAL ......87
RELATIONSHIP OF A STRAIGHT LINE AND CIRCUMFERENCE - TANGENTIAL ..88
RELATIONSHIP OF A STRAIGHT LINE AND CIRCUMFERENCE - SECANT .........89
RELATIVE POSITION OF TWO CIRCUMFERENCES - EXTERNAL .......................90
RELATIVE POSITION OF TWO CIRCUMFERENCES - INTERNAL .......................91
RELATIVE POSITION OF TWO CIRCUMFERENCES - EXTERNALLY TANGENT ... 92
RELATIVE POSITION OF TWO CIRCUMFERENCES - INTERNALLY TANGENT ... 93
RELATIVE POSITION OF TWO CIRCUMFERENCES - SECANT ...........................94
RELATIVE POSITION OF TWO CIRCUMFERENCES - CONCENTRIC ..................96
CHAPTER 2: STUDY OF CONGRUENCE
INTRODUCTION ..............................................................................................98
METAL INSET SQUARES ....................................................................................99
METAL INSET TRIANGLES ................................................................................ 103
CHAPTER 3: STUDY OF SIMILARITIES AND EQUIVALENCE
INTRODUCTION ............................................................................................ 106
PROPORTIONALITY WITH TRIANGLES - SENSORIAL PRESENTATION ................ 107
PROPORTIONALITY WITH RECTANGLES - SENSORIAL PRESENTATION ............. 109
PROPORTIONALITY WITH RECTANGLES - MATHEMATICAL PRESENTATION ...... 111
THE STUDY OF EQUIVALENCE: SENSORIAL PRESENTATION ............................. 114
CHAPTER 4: CONSTRUCTIVE TRIANGLES
INTRODUCTION ............................................................................................ 117
SEQUENCE OF ACTIVITIES .............................................................................. 118
THE TRIANGLE BOX ....................................................................................... 119
THE GREEN TRIANGLES FROM THE TRIANGLE BOX ........................................ 121
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Geometry Manual Volume 2
TRANSITIVE QUALITY OF EQUIVALENCIES ...................................................... 123
RELATIONSHIP OF SIDES OF THE TRIANGLE AND SIDES OF THE HALVES ......... 125
LARGE HEXAGON BOX - TRIANGLE AS THE POINT OF REFERENCE .................. 128
RELATIONSHIP OF SIDES OF TRIANGLE AND SIDES OF RHOMBUS .................. 131
LARGE HEXAGON BOX - HEXAGON AS THE POINT OF REFERENCE ................. 132
INSCRIBING A TRIANGLE IN A HEXAGON ....................................................... 133
CIRCUMSCRIBING A TRIANGLE BY A HEXAGON ............................................. 134
RELATIONSHIP OF LINES IN A HEXAGON CONSTRUCTED BY SIX TRIANGLES... 135
THE SMALL HEXAGON ................................................................................... 136
THE SMALL HEXAGON BOX ........................................................................... 138
TERMS OF THE DIFFERENCES AND THE RATIOS BETWEEN THE FIGURES .......... 142
RATIOS BETWEEN THE TWO HEXAGONS: H1 AND H2 .................................... 144
THE DIFFERENCE BETWEEN THE HEXAGONS, H1 AND H2 .............................. 146
EQUIVALENCE OF RED OBTUSE TRIANGLES AND RED EQUILATERAL TRIANGLE148
RATIO BETWEEN AN EQUILATERAL TRIANGLE AND ANOTHER
BUILT ON ITS HEIGHT ............................................................................... 151
EQUIVALENCE OF THE GREEN TRAPEZOID AND THE YELLOW TRIANGLE ........ 153
RELATIONSHIP BETWEEN INSCRIBED AND CIRCUMSCRIBED FIGURES ............. 155
THEOREM OF PYTHAGORAS .......................................................................... 157
FIRST EXTENSION OF THE THEOREM OF PYTHAGORAS .................................. 159
THE THEOREM OF PYTHAGORAS ................................................................... 165
FIRST INSET FOR THE THEOREM OF PYTHAGORAS ......................................... 170
SECOND INSET FOR THE THEOREM OF PYTHAGORAS .................................... 172
THIRD INSET FOR THE THEOREM OF PYTHAGORAS ....................................... 173
THE THEOREM OF PYTHAGORAS: THE CONSTRUCTION ................................ 177
CHAPTER 5: EQUIVALENCE
INTRODUCTION ............................................................................................ 181
EQUIVALENCE FROM THE TRIANGLE TO THE RECTANGLE .............................. 183
EQUIVALENCE FROM THE RHOMBUS TO THE RECTANGLE ............................. 185
EQUIVALENCE OF THE PARALLELOGRAM AND RECTANGLE ............................ 187
EQUIVALENCE OF THE TRAPEZOID AND RECTANGLE ..................................... 189
REGULAR POLYGONS .................................................................................... 194
STUDY OF APOTHEM ..................................................................................... 201
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