Table Of ContentCK-12 Geometry Concepts
Kathryn Dirga
Lori Jordan
Bill Zahner
Jim Sconyers
Victor Cifarelli
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Contents www.ck12.org
Contents
1 BasicsofGeometry 1
1.1 BasicGeometricDefinitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2 DistanceBetweenTwoPoints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.3 CongruentAnglesandAngleBisectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.4 MidpointsandSegmentBisectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
1.5 AngleMeasurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
1.6 AngleClassification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
1.7 ComplementaryAngles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
1.8 SupplementaryAngles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
1.9 LinearPairs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
1.10 VerticalAngles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
1.11 TriangleClassification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
1.12 PolygonClassification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
2 ReasoningandProof 49
2.1 ConjecturesandCounterexamples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
2.2 If-ThenStatements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
2.3 Converse,Inverse,andContrapositive . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
2.4 InductiveReasoningfromPatterns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
2.5 DeductiveReasoning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
2.6 TruthTables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
2.7 PropertiesofEqualityandCongruence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
2.8 Two-ColumnProofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
3 ParallelandPerpendicularLines 91
3.1 ParallelandSkewLines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
3.2 PerpendicularLines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
3.3 CorrespondingAngles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
3.4 AlternateInteriorAngles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
3.5 AlternateExteriorAngles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
3.6 SameSideInteriorAngles. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
3.7 SlopeintheCoordinatePlane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
3.8 ParallelLinesintheCoordinatePlane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
3.9 PerpendicularLinesintheCoordinatePlane . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
3.10 DistanceFormulaintheCoordinatePlane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
3.11 DistanceBetweenParallelLines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
4 TrianglesandCongruence 134
4.1 TriangleSumTheorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
4.2 ExteriorAnglesTheorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
4.3 CongruentTriangles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
4.4 CongruenceStatements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
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4.5 ThirdAngleTheorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152
4.6 SSSTriangleCongruence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
4.7 SASTriangleCongruence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162
4.8 ASAandAASTriangleCongruence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166
4.9 HLTriangleCongruence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171
4.10 IsoscelesTriangles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175
4.11 EquilateralTriangles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180
5 RelationshipswithTriangles 186
5.1 MidsegmentTheorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187
5.2 PerpendicularBisectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192
5.3 AngleBisectorsinTriangles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198
5.4 Medians . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203
5.5 Altitudes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208
5.6 ComparingAnglesandSidesinTriangles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211
5.7 TriangleInequalityTheorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215
5.8 IndirectProofinAlgebraandGeometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218
6 PolygonsandQuadrilaterals 223
6.1 InteriorAnglesinConvexPolygons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224
6.2 ExteriorAnglesinConvexPolygons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228
6.3 Parallelograms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232
6.4 QuadrilateralsthatareParallelograms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237
6.5 ParallelogramClassification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242
6.6 Trapezoids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247
6.7 Kites . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251
6.8 QuadrilateralClassification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255
7 Similarity 261
7.1 FormsofRatios . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262
7.2 ProportionProperties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 266
7.3 SimilarPolygonsandScaleFactors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271
7.4 AASimilarity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275
7.5 IndirectMeasurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279
7.6 SSSSimilarity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283
7.7 SASSimilarity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287
7.8 TriangleProportionality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291
7.9 ParallelLinesandTransversals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295
7.10 ProportionswithAngleBisectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 299
7.11 Dilation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303
7.12 DilationintheCoordinatePlane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 306
7.13 Self-Similarity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 310
8 RightTriangleTrigonometry 314
8.1 PythagoreanTheoremandPythagoreanTriples . . . . . . . . . . . . . . . . . . . . . . . . . . 315
8.2 ApplicationsofthePythagoreanTheorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 320
8.3 InscribedSimilarTriangles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 326
8.4 45-45-90RightTriangles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 331
8.5 30-60-90RightTriangles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 335
8.6 Sine,Cosine,Tangent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 339
8.7 TrigonometricRatioswithaCalculator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343
8.8 TrigonometryWordProblems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 347
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8.9 InverseTrigonometricRatios . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 351
8.10 LawsofSinesandCosines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 356
9 Circles 362
9.1 PartsofCircles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363
9.2 TangentLines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 367
9.3 ArcsinCircles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 372
9.4 ChordsinCircles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 376
9.5 InscribedAnglesinCircles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 381
9.6 InscribedQuadrilateralsinCircles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 385
9.7 AnglesOnandInsideaCircle. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 389
9.8 AnglesOutsideaCircle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393
9.9 SegmentsfromChords . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 397
9.10 SegmentsfromSecants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 401
9.11 SegmentsfromSecantsandTangents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 405
9.12 CirclesintheCoordinatePlane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 409
10 PerimeterandArea 414
10.1 AreaandPerimeterofRectangles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 415
10.2 AreaofaParallelogram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 418
10.3 AreaandPerimeterofTriangles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 421
10.4 AreaofCompositeShapes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 424
10.5 AreaandPerimeterofTrapezoids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 427
10.6 AreaandPerimeterofRhombusesandKites . . . . . . . . . . . . . . . . . . . . . . . . . . . . 430
10.7 AreaandPerimeterofSimilarPolygons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 435
10.8 Circumference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 439
10.9 ArcLength. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 443
10.10 AreaofaCircle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 447
10.11 AreaofSectorsandSegments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 450
10.12 AreaofRegularPolygons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 454
11 SurfaceAreaandVolume 460
11.1 Polyhedrons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 461
11.2 Cross-SectionsandNets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 465
11.3 Prisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 468
11.4 Cylinders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 473
11.5 Pyramids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 477
11.6 Cones . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 482
11.7 Spheres . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 487
11.8 CompositeSolids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 492
11.9 AreaandVolumeofSimilarSolids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 496
12 RigidTransformations 502
12.1 ReflectionSymmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 503
12.2 RotationSymmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 506
12.3 GeometricTranslations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 509
12.4 Rotations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 513
12.5 Reflections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 517
12.6 CompositionofTransformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 521
12.7 Tessellations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 526
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C 1
HAPTER
Basics of Geometry
Chapter Outline
1.1 BASIC GEOMETRIC DEFINITIONS
1.2 DISTANCE BETWEEN TWO POINTS
1.3 CONGRUENT ANGLES AND ANGLE BISECTORS
1.4 MIDPOINTS AND SEGMENT BISECTORS
1.5 ANGLE MEASUREMENT
1.6 ANGLE CLASSIFICATION
1.7 COMPLEMENTARY ANGLES
1.8 SUPPLEMENTARY ANGLES
1.9 LINEAR PAIRS
1.10 VERTICAL ANGLES
1.11 TRIANGLE CLASSIFICATION
1.12 POLYGON CLASSIFICATION
Introduction
In this chapter, students will learn about the building blocks of geometry. We will start with the basics: point, line
andplaneandbuilduponthoseterms. Fromhere, studentswilllearnaboutsegments, midpoints, angles, bisectors,
anglerelationships,andhowtoclassifypolygons.
1
1.1. BasicGeometricDefinitions www.ck12.org
1.1 Basic Geometric Definitions
Hereyou’lllearnthebasicgeometricdefinitionsandrulesyouwillneedtosucceedingeometry.
Whatifyouweregivenapictureofafigureoranobject,likeamapwithcitiesandroadsmarkedonit? Howcould
youexplainthatpicturegeometrically? AftercompletingthisConcept,you’llbeabletodescribesuchamapusing
geometricterms.
WatchThis
MEDIA
Clickimagetotheleftformorecontent.
CK-12Foundation: Chapter1BasicGeometricDefinitionsA
MEDIA
Clickimagetotheleftformorecontent.
JamesSousa:DefinitionsofandPostulatesInvolvingPoints,Lines,andPlanes
Guidance
A point is an exact location in space. A point describes a location, but has no size. Dots are used to represent
pointsinpicturesanddiagrams. Thesepointsaresaid“PointA,”“PointL”,and“PointF.”Pointsarelabeledwitha
CAPITALletter.
Alineisasetofinfinitelymanypointsthatextendforeverinbothdirections. Aline, likeapoint, doesnottakeup
space. Ithasdirection,locationandisalwaysstraight. Linesareone-dimensionalbecausetheyonlyhavelength(no
width). Alinecanbynamedoridentifiedusinganytwopointsonthatlineorwithalower-case,italicizedletter.
←→ ←→
ThislinecanbelabeledPQ, QPorjustg. Youwouldsay“linePQ,”“lineQP,”or“lineg,”respectively. Noticethat
←→ ←→
thelineoverthePQandQPhasarrowsoverboththePandQ. TheorderofPandQdoesnotmatter.
Aplaneisinfinitelymanyintersectinglinesthatextendforeverinalldirections. Thinkofaplaneasahugesheetof
paperthatgoesonforever. Planesareconsideredtobetwo-dimensionalbecausetheyhavealengthandawidth. A
planecanbeclassifiedbyanythreepointsintheplane.
ThisplanewouldbelabeledPlaneABCorPlaneM. Again,theorderofthelettersdoesnotmatter.
Wecanusepoint,line,andplanetodefinenewterms. Spaceisthesetofallpointsextendinginthreedimensions.
Think back to the plane. It extended along two different lines: up and down, and side to side. If we add a third
direction,wehavesomethingthatlookslikethree-dimensionalspace,orthereal-world.
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Pointsthatlieonthesamelinearecollinear. P,Q,R,S,andT arecollinearbecausetheyareallonlinew. Ifapoint
U werelocatedaboveorbelowlinew,itwouldbenon-collinear.
Points and/or lines within the same plane are coplanar. Lines h and i and points A,B,C,D,G, and K are coplanar
←→
inPlaneJ. LineKF andpointE arenon-coplanarwithPlaneJ.
Anendpointisapointattheendofalinesegment. Linesegmentsarelabeledbytheirendpoints,ABorBA. Notice
thatthebarovertheendpointshasNOarrows. Orderdoesnotmatter.
A ray is a part of a line with one endpoint that extends forever in the direction opposite that endpoint. A ray is
labeledbyitsendpointandoneotherpointontheline.
Oflines,linesegmentsandrays,raysaretheonlyonewhereordermatters. Whenlabeling,alwayswritetheendpoint
−→ ←−
underthesideWITHOUTthearrow,CDorDC.
Anintersectionisapointorsetofpointswherelines,planes,segments,orrayscrosseachother.
Postulates
Withthesenewdefinitions,wecanmakestatementsandgeneralizationsaboutthesegeometricfigures. Thissection
introduces a few basic postulates. Throughout this course we will be introducing Postulates and Theorems so it is
importantthatyouunderstandwhattheyareandhowtheydiffer.
Postulatesarebasicrulesofgeometry. Wecanassumethatallpostulatesaretrue,muchlikeadefinition. Theorems
arestatementsthatcanbeproventrueusingpostulates,definitions,andothertheoremsthathavealreadybeenproven.
Theonlydifferencebetweenatheoremandpostulateisthatapostulateisproventrue. Wewillprovetheoremslater
inthiscourse.
Postulate#1: Givenanytwodistinctpoints,thereisexactlyone(straight)linecontainingthosetwopoints.
Postulate#2: Givenanythreenon-collinearpoints,thereisexactlyoneplanecontainingthosethreepoints.
Postulate#3: Ifalineandaplanesharetwopoints,thentheentirelinelieswithintheplane.
Postulate#4: Iftwodistinctlinesintersect,theintersectionwillbeonepoint.
Postulate#5: Iftwodistinctplanesintersect,theintersectionwillbealine.
Whenmakinggeometricdrawings,besuretobeclearandlabelallpointsandlines.
ExampleA
WhatbestdescribesSanDiego,Californiaonaglobe?
A.point
B.line
C.plane
Answer: Acityisusuallylabeledwithadot,orpoint,onaglobe.
ExampleB
Usethepicturebelowtoanswerthesequestions.
a)ListanotherwaytolabelPlaneJ.
b)Listanotherwaytolabellineh.
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1.1. BasicGeometricDefinitions www.ck12.org
c)AreK andF collinear?
d)AreE,BandF coplanar?
Answer:
a)PlaneBDG. Anycombinationofthreecoplanarpointsthatarenotcollinearwouldbecorrect.
←→
b) AB. AnycombinationoftwoofthelettersA,B,orCwouldalsowork.
c)Yes
d)Yes
ExampleC
Describethepicturebelowusingallthegeometrictermsyouhavelearned.
Answer:
←→ ←→ ←→
AB andDarecoplanarinPlaneP,while BC and AC intersectatpointCwhichisnon-coplanar.
WatchthisvideoforhelpwiththeExamplesabove.
MEDIA
Clickimagetotheleftformorecontent.
CK-12Foundation: Chapter1BasicGeometricDefinitionsB
Vocabulary
Apointisanexactlocationinspace. Alineisinfinitelymanypointsthatextendforeverinbothdirections. Aplane
is infinitely many intersecting lines that extend forever in all directions. Space is the set of all points extending
in three dimensions. Points that lie on the same line are collinear. Points and/or lines within the same plane are
coplanar. Anendpointisapointattheendofpartofaline. Alinesegmentisapartofalinewithtwoendpoints. A
rayisapartofalinewithoneendpointthatextendsforeverinthedirectionoppositethatpoint. Anintersectionisa
pointorsetofpointswherelines,planes,segments,orrayscross. Apostulateisabasicruleofgeometryisassumed
to be true. A theorem is a statement that can be proven true using postulates, definitions, and other theorems that
havealreadybeenproven.
GuidedPractice
1. Whatbestdescribesthesurfaceofamoviescreen?
A.point
B.line
C.plane
2. Answerthefollowingquestionsaboutthepicture.
a)Islinel coplanarwithPlaneV,PlaneW,both,orneither?
b)AreRandQcollinear?
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