table of contentsunderstanding geometry table of con tens © 2012 the critical thinking co.™• ...
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UndersTandinggeomeTry TaBleofConTenTs
©2012TheCriticalThinkingCo.™•www.CriticalThinking.com•800-458-4849 iii
TABLE OF CONTENTSWhyUseThisBook..............................................................................................................iiTeachingSuggestions.........................................................................................................viAbouttheAuthor..................................................................................................................viStudentIntroduction...........................................................................................................viiDedication..........................................................................................................................viiiChapter 1 Fundamentals of Geometry Reasoning.................................... Pages 1-7
GeometryNotation....................................................................................1-2GeometryNotationPractice......................................................................... 3AllAboutLines............................................................................................. 4AllAboutPlanes........................................................................................... 5BuildIt!......................................................................................................... 6ThreePlanesinSpace................................................................................ 7
Chapter 2 Uncovering All the Angles....................................................... Pages 8-20TypesofAngles........................................................................................... 8AnglesActivity.............................................................................................. 9TriangleActivity.......................................................................................... 10TrianglePractice........................................................................................ 11QuadrilateralActivity.................................................................................. 12ComplementaryandSupplementaryAngles........................................13-14VerticalAngles........................................................................................... 15AnglesPuzzle1......................................................................................... 16AnglesPuzzle2......................................................................................... 17ParallelLinesandTransversalsI............................................................... 18ParallelLinesandTransversalsII............................................................. 19Puzzle3-The20-AngleProblem.............................................................. 20
Chapter 3 Triangle Properties................................................................. Pages 21-29SumofAngleMeasures............................................................................. 21ExteriorAngleExploration......................................................................... 22SumofTwoSides...................................................................................... 23TrianglePropertiesPractice....................................................................... 24TriangleOppositeProperty........................................................................ 25PropertiesofEquilateralandIsoscelesTriangles.................................26-27AlgebraandGeometry..........................................................................28-29
Chapter 4 The Pythagorean Theorem.................................................... Pages 30-41Exploration-WhatIsaTheorem?............................................................. 30PythagoreanTriples................................................................................... 31PythagoreanTriplesPractice..................................................................... 32UsingthePythagoreanTheorem......................................................... 33–34PythagoreanTheoremApplications......................................................35-36SolvingMulti-StepRightTriangleProblems.............................................. 37SpecialRightTriangles-AnIntroduction..............................................38-41
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TaBleofConTenTsUndersTandinggeomeTry
iv ©2012TheCriticalThinkingCo.™•www.CriticalThinking.com•800-458-4849
TABLE OF CONTENTS (Cont.)
Chapter 5 Uncovering All Polygons....................................................... Pages 42-51Polygons–InvestigationofAnglesandSides........................................... 42PolygonsAngleExploration..................................................................43-45SummaryofPolygonProperties................................................................ 46DiagonalExploration.............................................................................47-49TheHandshakeProblem......................................................................50-51
Chapter 6 Quadrilateral and Parallelogram Properties......................... Pages 52-78TheQuadrilateralFamily.......................................................................52-54VennDiagramActivity................................................................................ 55ParallelogramDiscovery.......................................................................56-57ParallelogramProperties.......................................................................58-59WorkingWithParallelograms................................................................60-63ExperimentingWithParallelograms......................................................64-66ALookatTrapezoids.............................................................................67-72Kite!Kite!Kite!............................................................................................ 73CriticalThinkingAboutQuadrilaterals........................................................ 74MidsegmentInvestigation.......................................................................... 75UsingAlgebratoSolveQuadrilateralProblems....................................76-77QuadrilateralMatching............................................................................... 78
Chapter 7 Metric Geometry..................................................................... Pages 79-96PerimeterandCircumference.................................................................... 79Pi(π)Investigation..................................................................................... 80PerimeterandCircumferenceActivity........................................................ 81Archimedes’IdeaforApproximatingPi.................................................82-83AreaofParallelograms..........................................................................84-85ParallelogramAreaActivity........................................................................ 86AreaofTrianglesandTrapezoids.........................................................87-89DiscoveringtheAreaofaTrapezoid.......................................................... 90AreaofaTrapezoid...............................................................................91-92AreaofCircles.......................................................................................93-95TheArbelosProblem.................................................................................. 96
Chapter 8 Geometric Constructions......................................................Pages 97-119AGeometricConstruction.......................................................................... 97HowtoBisectanAngle.........................................................................98-99HowtoCopyanAngle.............................................................................. 100HowtoConstructaPerpendicularBisectortoaLine........................101-103FindingtheMedianinaTriangle.......................................................104-105HowToConstructaLineParallelToAnotherLine................................... 106GeometricConstructionReview............................................................... 107Problem-SolvingWithGeometricConstructions............................... 108-117Problem-SolvingWithImpossibleConstructions...............................118-119
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UndersTandinggeomeTry TaBleofConTenTs
©2012TheCriticalThinkingCo.™•www.CriticalThinking.com•800-458-4849 v
Chapter 9 The Geometry of Three Dimensional Shapes.................. Pages 120-1343DShapes-Prisms..........................................................................120-122TheVolumeofCylinders...................................................................123-124VolumesofPyramidsandCones......................................................125-126TurnUptheVolume!.........................................................................127-128VolumeofaSphere...........................................................................129-130SurfaceAreaofPrisms............................................................................ 131SurfaceAreaofaCylinderActivity........................................................... 132FindingSurfaceArea................................................................................ 133Euler’sFormula........................................................................................ 134
Chapter 10 Symmetry and Transformations........................................ Pages 135-167WhatisVertical,Horizontal,andPointSymmetry?...........................135-137Transformations-Reflections...........................................................138-143Transformations-Translations..........................................................144-147Transformations-Rotations..............................................................148-153Transformations-Dilations...............................................................154-157GlideReflectionsandCompositions.................................................158-164Tessellations......................................................................................165-167
Chapter 11 Proving Triangles Congruent............................................ Pages 168-190IntroductiontoProofs-Congruency.................................................168-169SSSActivity.............................................................................................. 170SASActivity.............................................................................................. 171ASAandAASActivities............................................................................ 172SSAActivity.............................................................................................. 173TheEssenceofaGoodGeometricProof.........................................174-176Picture,Statement,andReason.......................................................177-178FindingCongruentTriangles.................................................................... 179TwoColumnProofs...........................................................................180-183InvestigateHypotenuse-LegTheorem............................................183-184SimilarFiguresandIntroductiontoSimilarityProofs........................185-187ProvingTrianglesSimilar..................................................................188-189TestYourReasoningSkills....................................................................... 190
Chapter 12 Coordinate Geometry......................................................... Pages 191-222WhatistheSlopeofaLine...................................................................... 191SlopeFormula...................................................................................192-196HowtoWriteanEquationofaLine...................................................197-201TheMidpointFormula.......................................................................202-205TheDistanceFormula.......................................................................206-210ReviewYourFormulas...................................................................... 211-212IntroductiontoCoordinateProofs.....................................................213-222
Glossary .............................................................................................. Pages 223-228
Answers .............................................................................................. Pages 229-263
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UndersTandinggeomeTry
Useyourthinking skillstofindthemissinganglesandrecordindegreesbelow�Figuresarenottoscale,sodonotmeasure�
Angles Puzzle 1
a_____ b_____ c_____ d_____ e_____ f_____ g_____
h_____ i_____ j_____ k_____ l_____ m_____
hg
f d
e
c
am
b
k
l
j
i
46°
83°
60°
UncoveringAlltheAngles
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78 ©2012TheCriticalThinkingCo.™•www.CriticalThinking.com•800-458-4849
UndersTandinggeomeTry QuadrilateralandParallelogramProperties
Quadrilateral Matching
1 Rectangle
______________
2 Rhombus
______________
3 Parallelogram
______________
4 Kite
______________
5 Trapezoid
______________
6 IsoscelesTrapezoid
______________
7 Square
______________
Matchthequadrilateralwithitsproperties�Youmayuseananswermorethanonce�
a Diagonalsarecongruent�
b Oppositesidesarecongruent�
c Oppositeanglesarecongruent�
d Consecutiveanglesaresupplementary�
e Diagonalsareperpendicular�
f Allsidesarecongruent�
g Allanglesarecongruent�
h Sumoftheanglemeasuresequals360°�
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©2012TheCriticalThinkingCo.™•www.CriticalThinking.com•800-458-4849 97
UndersTandinggeomeTry
Chapter 8 - Geometric Constructions
A Geometric Construction
GeometricConstructions
Ageometricconstructionisaconstructiondonewithonlyacompassandastraightedge�Ofcourse,agoodpencilisimportant�Inaclassicgeometricconstruction,youarenotallowedtomeasureorerase�Ifyoumakeamistake,it’sbestyoustartover�
Beforeweproblemsolveanddosomefunthinkingproblemswithconstructions,itisagoodideatoreviewhowtodotheseconstructions�
• Bisectinganangle�
• Copyinganangle�
• Constructingaperpendicularbisectortoaline�
• Constructingaperpendicularbisectortoalinefromapointonthenotontheline�
• Constructingalineparalleltoanotherlinethroughagivenpoint�
• Constructingsomepopularpolygons�
Forfun,createyourowndesignwithacompassandastraightedge�Trytofillupapageofpaperwithyourowndesign�
NoteMyconstructionsareactualdrawingssincetheyweredoneonthecomputer,butifyoufollowthedirectionsforeachofthese,youwillmakeperfectconstructions�
Geometric constructionsareusedbyengineersanddesigners�Theywillhelpyouseethebeautyofgeometry�
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©2012TheCriticalThinkingCo.™•www.CriticalThinking.com•800-458-4849 113
UndersTandinggeomeTry GeometricConstructions
Problem-Solving With Geometric Constructions (Cont.)
Plato(428-347 BC),afamousGreekphilosopherandmathematician,discoveredthatmanyconstructionscanbemadewithonlyacompassandnostraightedge�ThoseconstructionsarenowcalledMascheroni constructionsinhonorofItalianmathematicianLorenzoMascheroni(1750-1800)whowroteabookcalledThe Geometry of Compasses�
7 Usingonlyacompass,givensegmentAB,constructsegmentsthatare2,3,and4timesbiggerthanAB�
AncientGreekphilosopherdrawinghistheoreminthesand�
A B
8 IlostthecenterofMy Circle construction below�Ifyoudrawacircleandforgetwherethecenteris,drawtwonon-parallelchords�Constructtheperpendicularbisectorsofeachchordandwheretheperpendicularbisectorsmeetisyourmissingcenter�Tryit!
RememberYoucannotuseastraightedgesoyoucanonlyshowthepointsthatmarksuchdistances�
Whydoesthiswork?Explainyourthinking�_________________
____________________________________________________
____________________________________________________
HintOnceyoufindthecenter,maketwotriangles�Usethecenterasonepoint,andthepointswherethechordandthecircleintersecttohelpyouseewhatishappening�
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UndersTandinggeomeTry ProvingTrianglesCongruent
Picture, Statement, and Reason
Statement Reason
1
CA
B
D
a AB≅ CB
b ∠A≅ ∠C
a Given (Thefactisstatedinthepictureorintheinformation�)
b _______________
_______________
_______________
2
WR
T
V
a TV ≅ TV a _______________
_______________
_______________
3
1
2
a ∠1≅ ∠2 a _______________
_______________
_______________
Refertothepictureandwriteareasonforeachstatement�
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©2012TheCriticalThinkingCo.™•www.CriticalThinking.com•800-458-4849 181
UndersTandinggeomeTry
Two Column Proofs (Cont.)
ProvingTrianglesCongruent
Statement Reason
Proof3
R
BCA
2 4 3 1
Given:RC⊥ABandAC≅BC
Prove:∠1≅∠2
1 _______________
2 _______________
3 ∠ACRand∠BCR arerightangles�
4 _______________
5 RC ≅ RC
6 ∆ACR≅∆BCR
7 ∠3≅ ∠4
8 ∠1≅ ∠2
1 Given
2 Given
3 _______________
4 Allrightangles
arecongruent
5 _______________
6 _______________
7 _______________
8 _______________
Proof4
B
R
A
HLW
Given:WR≅HL,∠W≅∠H, ∠ALW≅∠BRH
Prove:∠∆AWL≅∆BHR and ∠A≅∠B
1 _______________
2 _______________
3 _______________
4 RL ≅ RL
5 WL≅ HR
6 ∠∆AWL≅∆BHR
7 ∠A≅ ∠B
1 Given
2 Given
3 Given
4 _______________
5 _______________
6 _______________
7 _______________
Refertothepictureandfillinthemissingreasonsorstatementsforeachproof�
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