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Patty Paper® Geometry Student Workbook • TB16988 • enasco.com/math PPG_SW_Interior_1st12 10/2/12 10:10 AM Page ii 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 16 15 14 13 12 Patty Paper® Geometry Student Workbook• TB16988 • enasco.com/math PPG Front/Intro_15th07 12/17/07 4:12 PM Page iii 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 TABLE OF CONTENTS iii Patty Paper® Geometry Student Workbook • TB16988 • enasco.com/math PPG Front/Intro_15th07 12/17/07 4:12 PM Page iv 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 Patty Paper® Geometry Student Workbook • TB16988 • enasco.com/math PPG Front/Intro_15th07 12/17/07 4:12 PM Page v 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 TABLE OF CONTENTS v Patty Paper® Geometry Student Workbook • TB16988 • enasco.com/math PPG Front/Intro_15th07 12/17/07 4:12 PM Page vi 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 Patty Paper® Geometry Student Workbook • TB16988 • enasco.com/math PPG Front/Intro_15th07 12/17/07 4:12 PM Page vii 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 Patty Paper® Geometry Student Workbook • TB16988 • enasco.com/math PPG Front/Intro_15th07 12/17/07 4:12 PM Page viii 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 Patty Paper® Geometry Student Workbook • TB16988 • enasco.com/math PPG Front/Intro_15th07 12/17/07 4:12 PM Page ix 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 Patty Paper® Geometry Student Workbook • TB16988 • enasco.com/math PPG Front/Intro_15th07 12/17/07 4:12 PM Page x 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 Patty Paper® Geometry Student Workbook • TB16988 • enasco.com/math PPG Front/Intro_15th07 12/17/07 4:12 PM Page 1 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 Patty Paper® Geometry Student Workbook • TB16988 • enasco.com/math PPG Front/Intro_15th07 12/17/07 4:12 PM Page 2 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 Patty Paper® Geometry Student Workbook • TB16988 • enasco.com/math PPG Front/Intro_15th07 12/17/07 4:12 PM Page 3 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 Patty Paper® Geometry Student Workbook • TB16988 • enasco.com/math PPG Front/Intro_15th07 12/17/07 4:12 PM Page 4 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 Patty Paper® Geometry Student Workbook • TB16988 • enasco.com/math PPG Front/Intro_15th07 12/17/07 4:12 PM Page 5 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 Patty Paper® Geometry Student Workbook • TB16988 • enasco.com/math PPG Front/Intro_15th07 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. _______________________________________________________________________________________ _______________________________________________________________________________________ _______________________________________________________________________________________ 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. _______________________________________________________________________________________ _______________________________________________________________________________________ _______________________________________________________________________________________ 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. _______________________________________________________________________________________ _______________________________________________________________________________________ _______________________________________________________________________________________ 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? _______________________________________________________________________________________ _______________________________________________________________________________________ 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