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DG4PSA 894 fm.qxd11/1/0611:16 AMPage iDiscoveringGeometryAn Investigative ApproachPractice Your Skillswith Answers
DG4PSA 894 fm.qxd4/28/087:53 PMPage iiTeacher’s Materials Project Editor: Elizabeth DeCarliProject Administrator: Brady GoldenCoordinating Writer: Jennifer North MorrisContributors: David Rasmussen, Ralph Bothe, Judy Hicks, Michael SerraAccuracy Checker: Dudley BrooksProduction Editor: Holly RudelitschCopyeditor: Jill PellarinEditorial Production Manager: Christine OsborneProduction Supervisor: Ann RothenbuhlerProduction Coordinator: Jennifer YoungText Designers: Jenny Somerville, Garry HarmanComposition, Technical Art, Prepress: ICC Macmillan Inc.Cover Designer: Jill KongabelPrinter: Data ReproductionsTextbook Product Manager: James RyanExecutive Editor: Casey FitzSimonsPublisher: Steven Rasmussen 2008 by Key Curriculum Press. All rights reserved.Cover Photo Credits: Background image: Doug Wilson/Westlight/Corbis.Construction site image: Sonda Dawes/The Image Works. All other images:Ken Karp Photography.Limited Reproduction PermissionThe publisher grants the teacher whose school has adopted Discovering Geometry,and who has received Discovering Geometry: An Investigative Approach, PracticeYour Skills with Answers as part of the Teaching Resources package for the book,the right to reproduce material for use in his or her own classroom. Unauthorizedcopying of Discovering Geometry: An Investigative Approach, Practice Your Skillswith Answers constitutes copyright infringement and is a violation of federal law. Key Curriculum Press is a registered trademark of Key Curriculum Press.All registered trademarks and trademarks in this book are the property oftheir respective holders.Key Curriculum Press1150 65th StreetEmeryville, CA s.comPrinted in the United States of America10 9 8 7 6 5 4 3 213 12 11 10 09 08ISBN 978-1-55953-894-7
DG4PSA 894 fm.qxd11/1/0611:16 AMPage iiiContentsIntroduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .viChapter 1Lesson 1.1:Lesson 1.2:Lesson 1.3:Lesson 1.4:Lesson 1.5:Lesson 1.6:Lesson 1.7:Lesson 1.8:Lesson 1.9:Building Blocks of Geometry . . . . . . .Poolroom Math . . . . . . . . . . . . . . . . . . . .What’s a Widget? . . . . . . . . . . . . . . . . . .Polygons . . . . . . . . . . . . . . . . . . . . . . . . . .Triangles . . . . . . . . . . . . . . . . . . . . . . . . . .Special Quadrilaterals . . . . . . . . . . . . . .Circles . . . . . . . . . . . . . . . . . . . . . . . . . . . .Space Geometry . . . . . . . . . . . . . . . . . . .A Picture Is Worth a Thousand Words. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .123456789Chapter 2Lesson 2.1:Lesson 2.2:Lesson 2.3:Lesson 2.4:Lesson 2.5:Lesson 2.6:Inductive Reasoning . . . . . . . . . .Finding the nth Term . . . . . . . . .Mathematical Modeling . . . . . . .Deductive Reasoning . . . . . . . . .Angle Relationships . . . . . . . . . . .Special Angles on Parallel Lines. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .101112131415Chapter 3Lesson 3.1:Lesson 3.2:Lesson 3.3:Lesson 3.4:Lesson 3.5:Lesson 3.6:Lesson 3.7:Lesson 3.8:Duplicating Segments and Angles . . .Constructing Perpendicular Bisectors .Constructing Perpendiculars to a LineConstructing Angle Bisectors . . . . . . . .Constructing Parallel Lines . . . . . . . . . .Construction Problems . . . . . . . . . . . . . .Constructing Points of Concurrency . .The Centroid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1617181920212223Chapter 4Lesson 4.1:Lesson 4.2:Lesson 4.3:Lesson 4.4:Lesson 4.5:Lesson 4.6:Lesson 4.7:Lesson 4.8:Triangle Sum Conjecture . . . . . . . . . . . . . . . . . .Properties of Isosceles Triangles . . . . . . . . . . .Triangle Inequalities . . . . . . . . . . . . . . . . . . . . . .Are There Congruence Shortcuts? . . . . . . . . . .Are There Other Congruence Shortcuts? . . . .Corresponding Parts of Congruent TrianglesFlowchart Thinking . . . . . . . . . . . . . . . . . . . . . . .Proving Special Triangle Conjectures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2425262728293031iii
DG4PSA 894 fm.qxd11/1/0611:16 AMPage ivChapter 5Lesson 5.1:Lesson 5.2:Lesson 5.3:Lesson 5.4:Lesson 5.5:Lesson 5.6:Lesson 5.7:Polygon Sum Conjecture . . . . . . . . . .Exterior Angles of a Polygon . . . . . .Kite and Trapezoid Properties . . . . .Properties of Midsegments . . . . . . . .Properties of Parallelograms . . . . . . .Properties of Special ParallelogramsProving Quadrilateral Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .32333435363738Chapter 6Lesson 6.1: Tangent Properties . . . . . . . . . . . . . . . . . . . . . . .Lesson 6.2: Chord Properties . . . . . . . . . . . . . . . . . . . . . . . . .Lesson 6.3: Arcs and Angles . . . . . . . . . . . . . . . . . . . . . . . . . .Lesson 6.4: Proving Circle Conjectures . . . . . . . . . . . . . . . . .Lesson 6.5: The Circumference/Diameter Ratio . . . . . . . . .Lesson 6.6: Around the World . . . . . . . . . . . . . . . . . . . . . . . .Lesson 6.7: Arc Length . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Exploration: Intersecting Secants, Tangents, and Chords. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3940414243444546Chapter 7Lesson 7.1: Transformations and Symmetry . . . .Lesson 7.2: Properties of Isometries . . . . . . . . . . .Lesson 7.3: Compositions of Transformations . .Lesson 7.4: Tessellations with Regular PolygonsLessons 7.5–7.8: Tessellations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .4748495051Chapter 8Lesson 8.1:Lesson 8.2:Lesson 8.3:Lesson 8.4:Lesson 8.5:Lesson 8.6:Lesson 8.7:Areas of Rectangles and ParallelogramsAreas of Triangles, Trapezoids, and KitesArea Problems . . . . . . . . . . . . . . . . . . . . . . . .Areas of Regular Polygons . . . . . . . . . . . . .Areas of Circles . . . . . . . . . . . . . . . . . . . . . . .Any Way You Slice It . . . . . . . . . . . . . . . . . .Surface Area . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .52535455565758Chapter 9Lesson 9.1:Lesson 9.2:Lesson 9.3:Lesson 9.4:Lesson 9.5:Lesson 9.6:ivThe Theorem of Pythagoras . . . . . . . . . . . . . .The Converse of the Pythagorean TheoremTwo Special Right Triangles . . . . . . . . . . . . . . .Story Problems . . . . . . . . . . . . . . . . . . . . . . . . . .Distance in Coordinate Geometry . . . . . . . . .Circles and the Pythagorean Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .596061626364
DG4PSA 894 fm.qxd11/1/0611:16 AMPage vChapter 10Lesson 10.1:Lesson 10.2:Lesson 10.3:Lesson 10.4:Lesson 10.5:Lesson 10.6:Lesson 10.7:The Geometry of Solids . . . . . . . .Volume of Prisms and CylindersVolume of Pyramids and Cones .Volume Problems . . . . . . . . . . . . . .Displacement and Density . . . . .Volume of a Sphere . . . . . . . . . . . .Surface Area of a Sphere . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .65666768697071Chapter 11Lesson 11.1:Lesson 11.2:Lesson 11.3:Lesson 11.4:Lesson 11.5:Lesson 11.6:Lesson 11.7:Similar Polygons . . . . . . . . . . . . . . . . . . . . . . . . . . . .Similar Triangles . . . . . . . . . . . . . . . . . . . . . . . . . . . .Indirect Measurement with Similar Triangles . .Corresponding Parts of Similar Triangles . . . . .Proportions with Area . . . . . . . . . . . . . . . . . . . . . .Proportions with Volume . . . . . . . . . . . . . . . . . . . .Proportional Segments Between Parallel Lines. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .72737475767778Chapter 12Lesson 12.1:Lesson 12.2:Lesson 12.3:Lesson 12.4:Lesson 12.5:Trigonometric Ratios . . . . . . . . . . . . . . . .Problem Solving with Right TrianglesThe Law of Sines . . . . . . . . . . . . . . . . . . .The Law of Cosines . . . . . . . . . . . . . . . . .Problem Solving with Trigonometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .7980818283Chapter 13Lesson 13.1:Lesson 13.2:Lesson 13.3:Lesson 13.4:Lesson 13.5:Lesson 13.6:Lesson 13.7:AnswersThe Premises of GeometryPlanning a Geometry ProofTriangle Proofs . . . . . . . . . .Quadrilateral Proofs . . . . . .Indirect Proof . . . . . . . . . . . .Circle Proofs . . . . . . . . . . . . .Similarity Proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .84858687888990. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .91. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .v
DG4PSA 894 fm.qxd11/1/0611:16 AMPage viIntroductionThe author and editors of Discovering Geometry: An Investigative Approach areaware of the importance of students developing geometry skills along withacquiring concepts through investigation. The student book includes manyskill-based exercises. These Practice Your Skills worksheets provide problemssimilar to the introductory exercises in each lesson of Discovering Geometry. Likethose exercises, these worksheets allow students to practice and reinforce theimportant procedures and skills developed in the lessons. Some of these problemsprovide non contextual skills practice. Others give students an opportunity toapply geometry concepts in fairly simple, straightforward contexts. Some are morecomplex problems that are broken down into small steps.You might assign the Practice Your Skills worksheet for every lesson, or only forthose lessons your students find particularly difficult. Or, you may wish toassign the worksheets on an individual basis, only to those students who needextra help. One worksheet has been provided for nearly every lesson. There areno worksheets for Chapter 0, and the optional tessellation lessons have beencombined into two worksheets. To save you the time and expense of copyingpages, you can give students the inexpensive Practice Your Skills Student Workbook,which does not have answers. Though the copyright allows you to copy pagesfrom Practice Your Skills with Answers for use with your students, the consumablePractice Your Skills Student Workbook should not be copied. Students, parents, andmentors can also download the student worksheets from www.keymath.com.vi
DG4PSA 894 01.qxd11/1/061:26 PMPage 1Lesson 1.1 Building Blocks of GeometryNamePeriodDate 3 cm.For Exercises 1–7, complete each statement. PSPN is .1. The midpoint of PQQS2. NQ . is .3. Another name for NS .4. S is the of SQ5. P is the midpoint of . .6. NS is .7. Another name for SN8. Name all pairs of congruent segments in KLMN. Use theN8 cmcongruence symbol to write your answer.KOM8 cm . D has coordinates (6, 1). Find the9. M( 4, 8) is the midpoint of DELcoordinates of E.For Exercises 10 and 11, use a ruler to draw each figure. Label thefigure and mark the congruent parts. and CD with M as the midpoint10. AB . C is the midpoint of AB , with and CD11. AB12. Sketch six points A, B, C, D, E, and F, no13. In the figure below, {B, C, H, E} is a set of and CD . AB 6.4 cmof both ABand CD 4.0 cm. A, B, and C arenot collinear.three of which are collinear. Name the linesdefined by these points. How many linesare there? , is theAC 1.5 cm. D, not on AB midpoint of AE , with AD 2BC.four coplanar points. Name two other setsof four coplanar points. How many sets offour coplanar points are there?BCADFEGHCubeDiscovering Geometry Practice Your Skills 2008 Key Curriculum PressCHAPTER 11
DG4PSA 894 01.qxd11/1/061:26 PMPage 2Lesson 1.2 Poolroom MathNamePeriodDateFor Exercises 1–5, use the figure at right to completeeach statement.B1. A is the of BAE.C is the o
14.11.2011 · Discovering Geometry Practice Your Skills CHAPTER 1 3 2008 Key Curriculum Press DG4PSA_894_01.qxd 11/1/06 1:26 PM Page 3. Lesson 1.4 Polygons Name Period Date For Exercises 1–8, complete the table. For Exercises 9 and 10, sketch and label each figure. Mark the congruences. Polygon name Number of sides Number of diagonals 1. Triangle 2. 2 3. 5 4. Hexagon 5. Heptagon 6.
