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Patty Paper Geometry Student Workbook
Author: Michael Serra
Editor: Crystal Mills
Editorial Assistance: Dan Bennett and Peter Rasmussen
Layout Design: Christy Butterfield
Production and Graphics: Ann Rothenbuhler and Crystal Mills
© 1994
®Patty
Michael Serra. All rights reserved.
Paper is a registered trademark of Michael Serra.
No part of this publication may be reproduced, stored in a retrieval system, or transmitted in
any form or by any means, electronic, photocopying, recording, or otherwise, without the prior
written permission of the publisher. Worksheets posted to an internet site, with or without
password protection, will be considered a violation of Federal Copright Law. Address inquiries
to the Permissions Dept. at the email below.
Playing It Smart
P.O. Box 27540
San Francisco, CA 94127
[email protected]
www.michaelserra.net
Printed in the U.S.
10 9 8 7 6 5 4 3 2 1
ISBN 978-1-55953-074-3
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TABLE OF CONTENTS
Note to the Teacher ..............................................................................................................vii
Note to the Student ...............................................................................................................ix
Historical Note on Geometric Constructions..........................................................................x
Introduction to Patty Paper Geometry: Basic Properties, Definitions, and Symbols ................1
Folding Properties...........................................................................................................1
Congruence Properties....................................................................................................2
The Terms and Symbols Used in Patty Paper Geometry ..................................................3
Investigation Set 1: Intersecting Lines .....................................................................................7
Guided Investigation 1.1: The Intersection of Two Lines ................................................7
Guided Investigation 1.2: Finding the Shortest Distance Between a Point and a Line......8
Guided Investigation 1.3: Vertical Angles .......................................................................9
Guided Investigation 1.4: Adjacent Angles and Linear Pairs .........................................10
Open Investigation 1.1: The Intersection of Two Lines .................................................11
Open Investigation 1.2: Finding the Shortest Distance Between a Point and a Line ......11
Open Investigation 1.3: Vertical Angles ........................................................................12
Open Investigation 1.4: Adjacent Angles and Linear Pairs............................................12
Investigation Set 2: Folding the Basic Geometric Constructions ...........................................15
Guided Investigation 2.1: Folding an Angle Bisector.....................................................17
Guided Investigation 2.2: Folding the Perpendicular Bisector of a Line Segment ..........19
Guided Investigation 2.3: Folding a Perpendicular from a Given Point
to a Given Line .....................................................................................................21
Guided Investigation 2.4: Folding a Perpendicular Through a Point on a Line ............21
Guided Investigation 2.5: Finding a Line Parallel to a Given Line
Through a Given Point .........................................................................................22
Open Investigation 2.1: Folding an Angle Bisector .......................................................23
Open Investigation 2.2: Folding the Perpendicular Bisector of a Line Segment .............24
Open Investigation 2.3: Folding a Perpendicular from a Given Point
to a Given Line ....................................................................................................25
Open Investigation 2.4: Folding a Perpendicular Through a Point on a Line ................26
Open Investigation 2.5: Finding a Line Parallel to a Given Line
Through a Given Point ........................................................................................27
Investigation Set 3: Special Points of Intersection .................................................................31
Guided Investigation 3.1: The Perpendicular Bisectors of the Sides of a Triangle..........32
Guided Investigation 3.2: The Angle Bisectors of a Triangle .........................................34
Guided Investigation 3.3: The Medians of a Triangle ...................................................36
Guided Investigation 3.4: The Altitudes of a Triangle...................................................37
Guided Investigation 3.5: The Circumcenter, Incenter, and Centroid of a Triangle .......39
Guided Investigation 3.6: The Euler Line......................................................................42
Open Investigation 3.1: The Perpendicular Bisectors of the Sides of a Triangle.............43
Open Investigation 3.2: The Angle Bisectors of a Triangle............................................45
Open Investigation 3.3: The Medians of a Triangle ......................................................46
Open Investigation 3.4: The Altitudes of a Triangle......................................................47
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Open Investigation 3.5: The Circumcenter, Incenter, and Centroid of a Triangle ..........49
Open Investigation 3.6: The Euler Line ........................................................................51
Investigation Set 4: Three Big Ideas......................................................................................55
Guided Investigation 4.1: Polygon Sum Conjectures.....................................................56
Guided Investigation 4.2: Isosceles Triangle Conjecture and its Converse.....................58
Guided Investigation 4.3: Parallel Lines Conjecture......................................................60
Guided Investigation 4.4: Converse of the Parallel Lines Conjecture ............................62
Open Investigation 4.1: Polygon Sum Conjectures........................................................63
Open Investigation 4.2: Isosceles Triangle Conjecture and its Converse........................65
Open Investigation 4.3: Parallel Lines Conjecture.........................................................67
Open Investigation 4.4: Converse of the Parallel Lines Conjecture ...............................69
Investigation Set 5: Midsegment Conjectures........................................................................73
Guided Investigation 5.1: Triangle Midsegment Conjectures ........................................74
Guided Investigation 5.2: Trapezoid Midsegment Conjectures .....................................76
Open Investigation 5.1: Triangle Midsegment Conjectures ...........................................79
Open Investigation 5.2: Trapezoid Midsegment Conjectures ........................................81
Investigation Set 6: Properties of Quadrilaterals ...................................................................87
Guided Investigation 6.1: Parallelogram Properties ......................................................88
Guided Investigation 6.2: Rhombus Properties .............................................................90
Guided Investigation 6.3: Rectangle Properties.............................................................92
Guided Investigation 6.4: Kite Properties......................................................................93
Open Investigation 6.1: Parallelogram Properties .........................................................94
Open Investigation 6.2: Rhombus Properties................................................................95
Open Investigation 6.3: Rectangle Properties................................................................96
Open Investigation 6.4: Kite Properties ........................................................................97
Investigation Set 7: Properties of Circles.............................................................................103
Guided Investigation 7.1: Finding the Center of a Circle ............................................105
Guided Investigation 7.2: Tangents to a Circle ...........................................................107
Guided Investigation 7.3: Tangent Segments to a Circle .............................................108
Guided Investigation 7.4: Central Angles, Arcs, and Chords ......................................109
Guided Investigation 7.5: Inscribed Angles .................................................................110
Guided Investigation 7.6: Angles Inscribed in the Same Arc .......................................111
Guided Investigation 7.7: Angles Inscribed in a Semicircle..........................................112
Guided Investigation 7.8: Parallel Lines Through a Circle ..........................................113
Guided Investigation 7.9: Cyclic Quadrilaterals..........................................................114
Open Investigation 7.1: Finding the Center of a Circle ...............................................115
Open Investigation 7.2: Tangents to a Circle ..............................................................116
Open Investigation 7.3: Tangent Segments to a Circle ................................................116
Open Investigation 7.4: Central Angles, Arcs, and Chords .........................................117
Open Investigation 7.5: Inscribed Angles....................................................................117
Open Investigation 7.6: Angles Inscribed in the Same Arc ..........................................118
iv PATTY PAPER GEOMETRY
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Open Investigation 7.7: Angles Inscribed in a Semicircle ............................................118
Open Investigation 7.8: Parallel Lines Through a Circle .............................................119
Open Investigation 7.9: Cyclic Quadrilaterals ............................................................119
Investigation Set 8: Congruent Triangles ............................................................................125
Guided Investigation 8.1: Side–Side–Side....................................................................126
Guided Investigation 8.2: Angle–Angle–Angle ............................................................127
Guided Investigation 8.3: Side–Angle–Side .................................................................128
Guided Investigation 8.4: Angle–Side–Angle...............................................................129
Guided Investigation 8.5: Side–Angle–Angle...............................................................130
Guided Investigation 8.6: Side–Side–Angle .................................................................131
Open Investigation 8.1: Side–Side–Side.......................................................................132
Open Investigation 8.2: Angle–Angle–Angle...............................................................133
Open Investigation 8.3: Side–Angle–Side ....................................................................134
Open Investigation 8.4: Angle–Side–Angle .................................................................135
Open Investigation 8.5: Side–Angle–Angle .................................................................136
Open Investigation 8.6: Side–Side–Angle ....................................................................137
Investigation Set 9: Transformations ..................................................................................145
Guided Investigation 9.1: Translations .......................................................................146
Guided Investigation 9.2: Rotations ...........................................................................148
Guided Investigation 9.3: Reflections .........................................................................150
Guided Investigation 9.4: Order of Points After Transformations...............................152
Guided Investigation 9.5: Two Reflections over Parallel Lines ....................................153
Guided Investigation 9.6: Two Reflections over Intersecting Lines .............................154
Open Investigation 9.1: Translations ..........................................................................155
Open Investigation 9.2: Rotations ..............................................................................156
Open Investigation 9.3: Reflections ............................................................................158
Open Investigation 9.4: Order of Points After Transformations .................................159
Open Investigation 9.5: Two Reflections over Parallel Lines.......................................160
Open Investigation 9.6: Two Reflections over Intersecting Lines ................................161
Investigation Set 10: Symmetry and Tessellations ...............................................................171
Guided Investigation 10.1: Reflectional Symmetry of Regular Polygons .....................174
Guided Investigation 10.2: Rotational Symmetry of Regular Polygons .......................175
Guided Investigation 10.3: Tessellations of Regular Polygons.....................................176
Guided Investigation 10.4: Tiling the Plane with Nonregular Polygons ......................178
Investigation 10.5: Creating Escher-Style Translation Tessellations.............................180
Investigation 10.6: Creating Escher-Style Rotation Tessellations.................................183
Investigation 10.7: Creating Escher-Style Glide Reflection Tessellations .....................185
Open Investigation 10.1: Reflectional Symmetry of Regular Polygons........................188
Open Investigation 10.2: Rotational Symmetry of Regular Polygons..........................189
Open Investigation 10.3: Tessellations of Regular Polygons .......................................190
Open Investigation 10.4: Tiling the Plane with Nonregular Polygons .........................191
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Investigation Set 11: Area...................................................................................................201
Guided Investigation 11.1: The Area Formula for Parallelograms ..............................202
Guided Investigation 11.2: The Area Formula for Triangles .......................................203
Guided Investigation 11.3: The Area Formula for Trapezoids ....................................204
Guided Investigation 11.4: The Area Formula for Circles...........................................205
Open Investigation 11.1: The Area Formula for Parallelograms .................................206
Open Investigation 11.2: The Area Formula for Triangles ..........................................206
Open Investigation 11.3: The Area Formula for Trapezoids .......................................207
Open Investigation 11.4: The Area Formula for Circles..............................................208
Investigation Set 12: The Theorem of Pythagoras ..............................................................217
Guided Investigation 12.1: The Pythagorean Theorem ...............................................219
Guided Investigation 12.2: The Pythagorean Theorem Revisited ................................221
Guided Investigation 12.3: The Return of the Pythagorean Theorem .........................222
Guided Investigation 12.4: The Converse of the Pythagorean Theorem......................223
Open Investigation 12.1: The Pythagorean Theorem ..................................................225
Open Investigation 12.2: The Pythagorean Theorem Revisited...................................227
Open Investigation 12.3: The Return of the Pythagorean Theorem ............................228
Open Investigation 12.4: The Converse of the Pythagorean Theorem.........................229
Conjecture List ...................................................................................................................235
Dot Grids ...........................................................................................................................240
Answer Key ........................................................................................................................247
vi PATTY PAPER GEOMETRY
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PATTY PAPER GEOMETRY
NOTE TO THE TEACHER
What are patty papers? Well, they are not papers named after some famous geometer
named Patty. Patty papers are the waxed squares of paper used by fast food restaurants to
separate hamburger patties. Gone is the slow and clumsy process of cutting up rolls of waxed
paper into conveniently sized squares the night before you do paper folding activities. Gone are
frustrating ink smears and messy mistakes. Patty papers eliminate all those hassles. You can write on
patty papers with pencils and felt-tip pens, and they come in a variety of sizes. The 5.5˝ or 6˝ squares
are the perfect size for students to use to discover geometric properties by folding and tracing. Look for
a supplier in the Yellow Pages under “Restaurant Equipment and Supplies” or order them directly from
Key Curriculum Press.
Patty paper geometry evolved rather quickly after I saw a teacher use patty papers in one paper folding
activity. It was exciting to see, and each time I worked with patty papers my excitement grew. The
students in my class were successful from the start. I found we could perform constructions more
accurately and make geometric discoveries faster with patty papers. Previously, when I asked my
students to construct the three angle bisectors of a triangle with compass and straightedge and then
asked them what they discovered, they would say that the three angle bisectors always intersect to form
a cute little triangle. WRONG! Discovering the four different points of concurrency in triangles was the
first set of investigations my students performed with patty papers, and the list of discoveries quickly
grew. Soon they were doing more than paper folding. They were using second sheets of patty papers to
check congruence of segments, angles, and polygons. They were able to transfer distances just as if they
were using a compass, and they used the right angled corners to quickly check for right angles or locate
perpendicular distances. Everything I needed for a full year of geometric discoveries was right there in a
box of patty papers!
When patty paper activities started appearing in my workshops for teachers, I devoted a brief five
minutes to them. Within months the number of patty paper activities grew until five minutes became
ten, then fifteen, then thirty minutes. I now do three-hour patty paper geometry workshops! Caution:
Patty paper geometry is addictive.
Cooperative learning is the most effective mode of classroom instruction for doing investigations in
Patty Paper Geometry. The best group structure for patty paper geometry is pair-share. In pair-share,
students who ordinarily work in groups of four break into two pairs. One student in each pair reads
the instructions to the partner while the partner does the folding. The pair compares its results with the
results of the other pair in the group. The group then makes its conjecture. For the next investigation,
both the pairs and the roles switch. This cooperative group structure will help you get all your students
involved in sharing the excitement of discovery learning. In pair-share, everyone has a role and shares
in the pride of discovery. Pair-share also helps reduce students’ math anxiety.
If you assign the investigations to students to work on individually or at home, they will need more
time, preparation, and guidance by the teacher before, during, and after each investigation. Students
will also miss an important opportunity to develop and practice good social skills.
NOTE TO THE TEACHER vii
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This blackline master book contains both guided and open investigations. The guided investigations
provide step-by-step instructions with illustrations. These investigations can be used as a teacher’s guide
or for those students who need more explanation and guidance. The open investigations are less
directed and encourage students to do more independent discovery. You may find you want to use the
guided investigations for some topics and the open investigations for other topics. At the end of each
investigation set there are exercise sets that reinforce concepts in the investigations and review
previously learned concepts. (A separate student workbook is available that contains the open
investigations and most of the exercise sets.)
In Investigation Set 10 there are three investigations that are not labeled Guided or Open. These
investigations give step-by-step instructions for creating Escher-style tessellations. Students will also
need tracing paper or light-weight typing paper to complete these investigations.
Ask each student to create a geometry notebook. The notebook should have four sections:
investigations, vocabulary, conjectures, and homework.
In the investigation section, students tape or staple the patty papers they used in their investigation to a
notebook page, then write a few sentences describing what they did and what they discovered in that
investigation.
In the vocabulary section, students keep a definition list of the geometric terms and symbols used. Each
definition should be a complete sentence with a picture illustrating the term being defined. Either the
teacher hands out this list or the students create their own lists.
In the conjecture section, students keep a list of the conjectures they make. Each conjecture, like the
definitions, should be a complete sentence with a picture illustrating the conjecture.
Students keep their homework and their class notes in a fourth section. (Use the exercise sets at the end
of each lesson for homework.)
Precede each Patty Paper Geometry lesson with a vocabulary review. Students then do their
investigations in their groups. After completing an investigation, the group makes its conjecture and
each students adds it to his or her conjecture list. Finally, students work the exercises at the end of the
lesson. In the exercises, students practice their geometry vocabulary, perform patty paper geometry
constructions, and solve geometry problems based on their discoveries. In this way they gain a deeper
understanding and appreciation of the discoveries just made.
Patty Paper Geometry was designed as a resource to supplement high school geometry courses or as a
text in pre-geometry, pre-algebra, and middle grades mathematics programs. An investigation set may
take two to three days, and the exercise sets may take an additional two to three days. Time will vary
depending on age, abilities, and how well students work cooperatively.
Michael Serra
viii PATTY PAPER GEOMETRY
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NOTE TO THE STUDENT
Patty papers are thin, lightly waxed 5.5˝ or 6˝ squares of paper that you
are going to use to make geometric discoveries. (Fast food restaurants
use patty papers between the uncooked hamburger patties.) Patty Paper
Geometry was designed for you to “learn by doing.” You are going to
be asked to read and follow the steps to a number of geometric
investigations using patty papers. These investigations should lead you
to discover most of the properties of geometric figures that are studied
in high school geometry. For example: What do we know about
isosceles triangles? What happens when a line intersects a pair of
parallel lines? What is true about the diagonals of a rhombus?
Geometric properties are very important because we use geometry to
help us understand the way things work. We can calculate perimeters,
areas, and volumes only with the help of geometry. With geometry we
can build bridges, skyscrapers, and airplanes. With geometry we can
begin to understand why elephants have big ears or why grasshoppers
can jump many times their height! With Patty Paper Geometry you will
get a hands-on introduction to geometry.
You should keep a geometry notebook to organize all the investigations
you do, the vocabulary you use, the conjectures you make, and the
exercises and homework you complete. This will help you retain what
you learn and will be your reference as you base new discoveries on
what you’ve learned before.
What kinds of geometric properties can you discover given a supply of
patty papers? Actually, any property that can be discovered using a
compass and a straightedge can be discovered by folding patty papers.
Tracing segments, angles, and polygons and using patty papers to
compare lengths and angle measurements will become very useful
methods of discovery for you. In Patty Paper Geometry you will
discover most of the properties of high school geometry. What follows
in Patty Paper Geometry are investigations leading to geometric
discoveries followed by exercises in which you can apply your
discoveries. Have fun!
Michael Serra
NOTE TO THE STUDENT ix
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HISTORICAL NOTE ON GEOMETRIC
CONSTRUCTIONS
For centuries, Babylonian and Egyptian mathematical writings were very
“cookbook” in style. That is, when some new mathematics was discovered there
was no attempt to explain why it was true—the reader was just told, “do thus and
so . . .” This early mathematics was very practical and dealt primarily with rules for
finding measurements of lengths, areas, and volumes.
The Iron Age ushered in new and improved tools. As trade developed in the
Phoenician and Greek regions of the Mediterranean, it brought with it the
development of coins, the alphabet, and a love and respect for clear, rational
thought. The philosophers and mathematicians were among the highest classes of
this ancient Greek society. For the first time, mathematicians were asking
fundamental questions: “Why are these rules that were handed down from the
earlier Egyptian and Babylonian mathematicians true?”
As geometry moved from the earlier, practical Egyptian stage to the logical Greek
stage, it became a passionate game with geometers to see what they could create
using just a compass and straightedge (a geometric construction). Later, other
geometers discovered many new geometric properties while continuing to play this
construction game. Geometers puzzled over a number of construction problems and
later found them to be impossible using just a compass and straightedge. Perhaps
one of the most famous impossible constructions is the trisection of an angle. The
angle trisection problem asks, “Given an arbitrary angle, is it possible using
compass and straightedge alone to divide the angle into three congruent parts?” The
answer turns out to be no. Over the centuries, geometers have explored the game
with compass alone or straightedge alone. They also discovered that all the
constructions with compass and straightedge can be performed using only a doubleedged straightedge. Patty paper geometry is yet another variation on this game that
has been played for well over 2000 years. (After you’ve gotten the hang of patty
paper constructions, you might see if you can use patty papers to trisect an angle!)
x PATTY PAPER GEOMETRY
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INTRODUCTION TO PATTY PAPER GEOMETRY
BASIC PROPERTIES, DEFINITIONS, AND
SYMBOLS
In Patty Paper Geometry you are going to discover geometric properties using patty papers instead of a
compass and straightedge. First you will have to assume a few basic folding and congruence properties.
Also, you need to agree on some basic definitions and become familiar with commonly used geometric
symbols and notation. Carefully read this section and do the exercises before starting the investigations.
FOLDING PROPERTIES
FP-1: It is possible to fold a patty paper so that the
crease forms a line. This fold shows that the
intersection of two planes is a line.
one plane
a line
another plane
FP-2: If you draw two points on your patty paper, it
is possible to fold the paper so that the crease (line)
passes through the two given points. This property
shows that two points determine a line.
FP-3: Patty paper can be folded so that a point on
the paper can be placed over another point on the
same paper.
FP-4: Patty paper can be folded so that a line (or a
portion of the line) on the paper can be placed over
another line on the same paper.
BASIC PROPERTIES, DEFINITIONS, AND SYMBOLS 1
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CONGRUENCE PROPERTIES
Definition: Two geometric figures are congruent if they have the same size and shape. If you place
one of the figures on top of the other figure, then they will match exactly. The symbol for congruence is ≅.
With the ability to trace segments and angles from one patty paper to another, it is possible to duplicate
segments, angles, and polygons. This also gives you the ability to check to see if two different segments
are congruent or if two different angles are congruent. Here are the congruence properties of patty
paper geometry.
CP-1: A segment can be constructed congruent to
another segment on a patty paper by tracing the
original segment onto a second patty paper.
CP-2: Two segments are congruent if one segment
can be placed exactly on top of the other by folding
or if it can be traced onto another patty paper and
the copy placed exactly on top of the other segment.
CP-3: An angle can be constructed congruent to
another angle by placing a second patty paper over
the angle and tracing the original angle onto the
second patty paper.
CP-4: Two angles are congruent if one angle can be
placed exactly on top of the other by folding or if it
can be traced onto another patty paper and the copy
placed exactly on top of the other angle.
2 INTRODUCTION
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THE TERMS AND SYMBOLS USED IN PATTY PAPER GEOMETRY
C
B
B
A
A
Line AB
↔
AB
A
B
A
Ray AB
→
AB
B
Segment AB

AB
Angle ABC
∠ABC
When you compare the sizes (lengths or areas) of geometric figures, you will say the measures are equal
or not equal. When you compare the shapes of geometric figures, you will say the figures are congruent
or not congruent. When you label geometric figures, use similar slashes to mark segments that have the
same length or angles that have the same degree measure.
A
C
D
B
A
B
C
The length of segment AB is equal to the length
of segment BC (AB = BC).
The measure of angle ABD is equal to the
measure of angle DBC (m∠ABD = m∠DBC).
Line segment AB is congruent to


line segment BC (AB ≅ BC).
Angle ABD is congruent to
angle DBC (∠ABD ≅ ∠DBC).
Two polygons are congruent if and only if they have all their corresponding angles congruent and all
their corresponding sides congruent. The order of the letters in the statement of congruence tells you
which segments and angles are corresponding and congruent.
Y
B
L
O
G

R





If ∆BOY ≅ ∆GRL, then ∠B ≅ ∠G, ∠O ≅ ∠R, ∠Y ≅ ∠L, BO ≅ GR, OY ≅ RL, and BY ≅ GL.
BASIC PROPERTIES, DEFINITIONS, AND SYMBOLS 3
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When you wish to label lines or line segments as
parallel, you mark the lines with similar arrow
slashes. (The symbol for parallel is //.)
When you wish to label lines as perpendicular,
you mark the intersection with a small box to
indicate the right angle. (The symbol for
perpendicular is ⊥.)
B
C
D
A
l1
I
G
C
E
J
D
H
F
↔
↔
AB // CD
↔
↔
↔
EF // GH // IJ
↔
l1 ⊥ CD
4 INTRODUCTION
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EXERCISE SET 0A
Your first task is to convince yourself that you can indeed perform the above construction properties.
1. Draw two points on a patty paper and fold so that the crease passes through the two points. (FP-2)
2. Draw two points on a patty paper and fold so that the two points lie on top of each other. (FP-3)
3. Draw two lines on a patty paper so that the lines go all the way to the ends of the patty paper and
do not cross each other on the patty paper. Fold the patty paper so that the lines (or portions of the
lines) lie on top of each other. (FP-4)
4. Draw two lines on a patty paper so that the lines go all the way to the ends of the patty paper and
cross each other on the patty paper. Fold the patty paper so that the lines (or portions of the lines)
lie on top of each other. (FP-4)
5. Draw a segment on a patty paper. Place a second patty paper over it and trace the original segment.
(CP-1)
6. Draw an angle on a patty paper. Place a second patty paper over it and trace the original angle.
(CP-3)
EXERCISE SET 0B
To review basic geometry vocabulary, match each term with the diagram that best illustrates it.
1.
___________ point
2.
___________ line
3.
___________ plane
4.
___________ ray
5.
___________ segment
6.
___________ midpoint
7.
___________ triangle
8.
___________ acute angle
9.
___________ obtuse angle
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
10. ___________ right angle
11. ___________ parallel lines
12. ___________ perpendicular lines
13. ___________ compass
14. ___________ straightedge
15. ___________ ruler
16. ___________ protractor
BASIC PROPERTIES, DEFINITIONS, AND SYMBOLS 5
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12/17/07
4:12 PM
Page 6
EXERCISE SET 0C
Sketch a diagram that best illustrates each term.
1. Polygon
2. Triangle
3. Acute Triangle
4. Obtuse Triangle
5. Scalene Triangle
6. Isosceles Triangle
7. Equilateral Triangle
8. Right Triangle
9. Quadrilateral
10. Trapezoid
11. Kite
12. Parallelogram
13. Rhombus
14. Rectangle
15. Square
6 INTRODUCTION
Patty Paper® Geometry Student Workbook • TB16988 • enasco.com/math
PPG_SW_Interior_17th11 1/6/11 4:22 PM Page 6
Open Investigation 2.3
FOLDING A PERPENDICULAR FROM A GIVEN POINT TO A GIVEN LINE
Step 1:
Fold or draw a line on a patty paper.
Place a dot on your patty paper to
represent the given point.
Step 2:
Fold your patty paper so that the crease passes through the given point and is perpendicular
to the given line. You may need to experiment a couple of times to find the perpendicular,
but you can do it!
Step 3:
Use a corner of another patty paper to check if the angles formed by the crease and the given
line are right angles.
Describe the method you used to fold a perpendicular from a given point to a given line.
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This construction allows you to fold an altitude. It is also a way to determine the shortest distance from
a point to a line.
Open Investigation 2.4
FOLDING A PERPENDICULAR THROUGH A POINT ON A LINE
Step 1:
Start with a line and a point on the line.
Experiment to find out how you would fold
to construct a line perpendicular to the given
line passing through the given point.
Step 2:
Use a corner of another patty paper to check if the angles formed by the crease and the given
line are right angles.
Describe the method you used to fold a line perpendicular to the given line and through the
given point.
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This construction is very useful because it allows you to fold a right angle. Many polygons contain
right angles. Look around! Right angles are used more than any other angle.
6 PATTY PAPER GEOMETRY
Patty Paper® Geometry Student Workbook • TB16988 • enasco.com/math
PPG_SW_Interior_17th11 1/6/11 4:22 PM Page 7
Open Investigation 2.5
FINDING A LINE PARALLEL TO A GIVEN LINE THROUGH A GIVEN POINT
There are several ways to do this construction. When you’re finished, compare your method to
methods used by others in the class.
Step 1:
Start with a line and a point not on the line.
Step 2:
Discover a method for folding a line through the point so that the line is parallel to the
given line.
Describe the method you used to construct the parallel line.
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Describe what happens when you have two lines that are perpendicular to a third line and all of the
lines on the same plane. What is the relationship between the two lines?
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What you learned in this investigation will allow you to fold to create parallelograms, rectangles, and
other figures with parallel lines. Try folding a parallelogram on your patty paper. You will discover
properties of parallelograms in a later lesson.
Patty Paper® Geometry Student Workbook • TB16988 • enasco.com/math
STUDENT WORKBOOK 7