CK-12 Geometry Concepts Kathryn Dirga Lori Jordan Bill Zahner Jim Sconyers Victor Cifarelli SayThankstotheAuthors Clickhttp://www.ck12.org/saythanks (Nosigninrequired) www.ck12.org AUTHORS KathrynDirga To access a customizable version of this book, as well as other LoriJordan interactivecontent,visitwww.ck12.org BillZahner JimSconyers VictorCifarelli CK-12 Foundation is a non-profit organization with a mission to reduce the cost of textbook materials for the K-12 market both in the U.S. and worldwide. Using an open-content, web-based collaborative model termed the FlexBook®, CK-12 intends to pioneerthegenerationanddistributionofhigh-qualityeducational content that will serve both as core text as well as provide an adaptiveenvironmentforlearning,poweredthroughtheFlexBook Platform®. 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Printed: September29,2012 iii 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 iv www.ck12.org Contents 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 v Contents www.ck12.org 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 vi www.ck12.org Chapter1. BasicsofGeometry 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. 2 www.ck12.org Chapter1. BasicsofGeometry 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. 3 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? 4