Download Ruler Postulate: The points on a line can be matched one to one

RulerPostulate:Thepointsonalinecanbematchedonetoonewiththe
Point,Point,Line,andPlanePostulates:
REALnumbers.TheREALnumberthatcorrespondstoapointisthe
Postulate5:ThroughanytwopointsthereexistsexactlyONEline.
COORDINATEofthepoint.TheDISTANCEbetweenpointsAandB,written
Postulate6:AlinecontainsatleastTWOpoints.
asAB,istheabsolutevalueofthedifferenceofthecoordinatesofAandB.
Postulate7:IfTWOlinesintersect,thentheirintersectionisexactlyonePOINT.
Postulate8:
SegmentAdditionPostulate:IfBisBETWEENAandC,then
ThroughanythreenoncollinearpointsthereexistsexactlyonePLANE.
AB+BC=AC.IfAB+BC=AC,thenBisBETWEENAandC.
Postulate9:
AplanecontainsatleastthreeNONCOLLINEARpoints.
MidpointFormula:Themidpointofasegmentisthepointhalfwayacrossthe
Postulate10:IfTWOpointslieinaplane,thentheLINEcontainingthemliesin
segment.
theplane.
Themidpointofasegmentinthex-ycoordinateplaneisfoundbyaveragingthe
Postulate11:IfTWOplanesintersect,thentheirintersectionisaLINE.
twox-valuesandaveragingthetwoy-values.
AlgebraicPropertiesofEquality
AdditionProperty:
Ifa=b,thena+c=b+c.
Themidpointofasegmentwithendpoints(X1,Y1)and(X2,Y2)inthex-y
coordinateplaneis…
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SubtractionProperty:
ProtractorPostulate:Theraysofanyanglecanbematchedonetoonewiththe
realnumbersfrom0to360.
MultiplicationProperty: Ifa=b,thenaxc=bxc.
DivisionProperty:
Ifa=b,thena/c=b/c.
SubstitutionProperty:
Ifa=b,thenacanbesubstitutedforbinanyequationorexpression.
AngleAdditionPostulate:
If∠𝐴𝐵𝐶isadjacentto∠𝐷𝐵𝐶then𝑚∠𝐴𝐵𝐶 + 𝑚∠𝐷𝐵𝐶 = 𝑚∠𝑨𝑩𝑫.
TheDistributiveProperty:
a(b+c)=(b+c)a=ab+ac,wherea,b,andcarerealnumbers.
DistanceFormula:
Thedistanceorlength,AB,of𝐴𝐵isthetotallengthacrossthesegment.
ReflexiveProperty:Foranyrealnumbera,a=a.
Thedistance(d)orlengthofasegmentwithendpoints(X1,Y1)and
(X2,Y2)inthex-ycoordinateplaneis… 𝑑 =
Ifa=b,thena-c=b-c.
(𝑋1 − 𝑋2)! + (𝑌1 − 𝑌2)! SymmetricProperty:Foranyrealnumbersaandb,ifa=b,thenb=a.
TransitiveProperty
Foranyrealnumbersa,b,andc,ifa=bandb=c,thena=c.
Theorem2.1CongruenceofSegments:
Theorem3.1:AlternateInteriorAnglesTheorem
Segmentcongruenceisreflexive,symmetric,andtransitive.
Iftwoparallellinesarecutbyatransversal,thenthepairs
ofALTERNATEinterioranglesareCONGRUENT.
Theorem2.2CongruenceofAngles
Anglecongruenceisreflexive,symmetric,andtransitive.
Theorem3.2:AlternateExteriorAnglesTheorem
IfTWOparallellinesareCUTbyatransversal,thenthePAIRS
Theorem2.3:RightAnglesCongruenceTheorem
ofalternateEXTERIORanglesarecongruent.
AllrightanglesareCONGRUENT.
Theorem3.3:ConsecutiveInteriorAnglesTheoremIftwoparallellinesarecut
byatransversal,thenthepairsofCONSECUTIVEinterioranglesare
SUPPLEMENTARY.
Theorem2.5:CongruentSupplementsTheoremIf∠1and∠2aresupplementary
and∠2and∠3aresupplementary,then∠𝟏 ≅ ∠𝟑.
Postulate12:LinearPairPostulateIftwoanglesformalinearpair,thenthetwo
anglesareSUPPLEMENTARY.
Theorem2.6:VerticalAnglesCongruenceTheorem
If∠1and∠2areverticalangles,then∠𝟏 ≅ ∠𝟐.
Postulate13:ParallelPostulate
IfthereisaLINEandaPOINTnotontheline,thenthereisexactlyONEline
throughthepointparallel(∥)tothegivenline.
Postulate14:PerpendicularPostulate
IfthereisaLINEandaPOINTnotontheline,thenthereisexactlyONEline
throughthepointperpendicular(⊥)tothegivenline.
Postulate15:CorrespondingAnglesPostulate
IftwoPARALLELlinesarecutbyaTRANSVERSAL
thenthepairsofCORRESPONDINGanglesarecongruent.
Postulate16:CorrespondingAnglesConverseIftwolinesarecutbytransversal
sothepairsofcorrespondinganglesareCONGRUENT,thenthelinesare
PARALLEL.
Theorem3.4:AlternateInteriorAnglesConverse
Iftwolinesarecutbytransversalsothepairsofalternateinteriorangles
areCONGRUENT,thenthelinesarePARALLEL
Theorem3.5:AlternateExteriorAnglesConverse
Iftwolinesarecutbytransversalsothepairsofalternateexteriorangles
areCONGRUENT,thenthelinesarePARALLEL.
Theorem3.6:ConsecutiveInteriorAnglesConverse
Iftwolinesarecutbytransversalsothepairsofconsecutiveinterioranglesare
SUPPLEMENTARY,thenthelinesarePARALLEL.
Theorem3.7:TransitivePropertyofParallelLines
Iftwolinesareparalleltothesameline,thentheyarePARALLELtoeachother.
Theorem3.8:
Theorem4.3:ThirdAnglesTheorem
Iftwolinesintersecttoformalinearpairofcongruentangles,
IFTWOANGLESOFONETRIANGLEARECONGRUENTTOTHETWOANGLESOFA
thenthelinesarePERPENDICULAR.
SECONDTRIANGLE,THENTHETHIRDANGLEOFTHEFIRSTTRIANGLEIS
CONGRUENTTOTHETHIRDANGLEOFTHESECONDTRIANGLE.
Theorem3.9:Iftwolinesareperpendicular,thentheyintersect
Theorem4.4:PropertiesofCongruentTriangles
toformFOURRIGHTANGLES.
Theorem3.10:Iftwooftwoadjacentacuteanglesareperpendicular,
thentheanglesareCOMPLEMENTARY.
Theorem3.11:PerpendicularTransversalTheorem
Ifatransversalisperpendiculartooneoftwoparallellines,then
itisPERPENDICULARtotheotherline.
Theorem3.12:LinesPerpendiculartoaTransversalTheorem
Inaplane,iftwolinesareperpendiculartothesameline,then
theyarePARALLELtoeachother.
Theorem4.1:TriangleSumTheorem
THEANGLESOFATRIANGLEADDUPTO180°.
CorollarytotheTriangleSumTheorem
THENON-RIGHTANGLESOFARIGHTTRIANGLEARECOMPLEMENTARY.
Theorem4.2:ExteriorAngleTheorem
THEMEASUREOFANEXTERIORANGLEOFATRIANGLEISEQUALTOTHESUMOF
THETWONON-ADJACENTINTERIORANGLES.
ReflexivePropertyofCongruentTriangles:∆𝐴𝐵𝐶 ≅ ∆𝐴𝐵𝐶
SymmetricPropertyofCongruentTriangles
IF∆𝐴𝐵𝐶 ≅ ∆𝐷𝐸𝐹,THEN∆𝐷𝐸𝐹 ≅ ∆𝐴𝐵𝐶.
TransitivePropertyofCongruentTriangles
IF∆𝐴𝐵𝐶 ≅ ∆𝐷𝐸𝐹AND∆𝐷𝐸𝐹 ≅ ∆𝐺𝐻𝐼,THEN∆𝐴𝐵𝐶 ≅ ∆𝐺𝐻𝐼.
Postulate19:Side-Side-Side(SSS)CongruencePostulate
IfTHREEsidesofonetrianglearecongruenttoTHREEsidesofasecondtriangle,
thenthetwotrianglesareCONGRUENT.
Postulate20:Side-Angle-Side(SAS)CongruencePostulate
IFTWOSIDESANDTHEINCLUDEDANGLEOFONETRIANGLEARECONGRUENTTO
TWOSIDESANDTHEINCLUDEDANGLEOFASECONDTRIANGLE,THENTHETWO
TRIANGLESARECONGRUENT.
Theorem4.5:Hypotenuse-Leg(HL)CongruenceTheorem
IFTHEHYPOTENUSEANDALEGOFONERIGHTTRIANGLEARECONGRUENTTO
THEHYPOTENUSEANDALEGOFASECONDTRIANGLE,THENTHETWO
TRIANGLESARECONGRUENT.
Postulate21:Angle-Side-Angle(ASA)CongruencePostulate
Theorem9.1:Translation(Slide)Theorem Atranslationisanisometry.
IFTWOANGLESANDTHEINCLUDEDSIDEOFONETRIANGLEARECONGRUENTTO
TWOANGLESANDTHEINCLUDEDSIDEOFASECONDTRIANGLE,THENTHETWO
Theorem9.2:Reflection(Flip)TheoremAreflectionisanisometry.
TRIANGLESARECONGRUENT.
CoordinateRulesforReflections
Ifpreimage(x,y)isreflectedoverthex-axis(y=0)…itsimageis(x,-y). Theorem4.6:Angle-Angle-Side(AAS)CongruenceTheorem
IFTWOANGLESANDTHENON-INCLUDEDSIDEOFONETRIANGLEARE
CONGRUENTTOTWOANGLESANDTHECORRESPONDINGNON-INCLUDEDSIDE
OFASECONDTRIANGLE,THENTHETWOTRIANGLESARECONGRUENT.
CorrespondingPartsofCongruentTrianglesareCongruent(CPCTC)
IFTHEREARETWOORMORETRIANGLESTHATARECONGRUENT,THENTHEIR
CORRESPONDINGPARTS(SIDESANDANGLES)ARECONGRUENT.
Theorem4.7:BaseAnglesTheorem
Iftwosidesofatrianglearecongruent,thenTHEANGLESOPPOSITETHEMARE
CONGRUENT.
Theorem4.8:ConverseofBaseAnglesTheorem
Iftwoanglesofatrianglearecongruent,thenTHESIDESOPPOSITETHEMARE
CONGRUENT.
CorollarytotheBaseAnglesTheorem
Ifthreesidesofatrianglearecongruent(equilateral),thenITISEQUIANGULAR.
CorollarytotheConverseofBaseAnglesTheorem
Ifthreeanglesofatrianglearecongruent(equiangular),thenITISEQUILATERAL.
Ifpreimage(x,y)isreflectedoverthey-axis(x=0)…itsimageis(-x,y). Ifpreimage(x,y)isreflectedoverthey=x…itsimageis(y,x). Ifpreimage(x,y)isreflectedoverthey=-x…itsimageis(-y,-x).
Theorem9.3:Rotation(Spin)Theorem
Arotationisanisometry.
CoordinateRulesforRotations
Whenapoint(a,b)isrotatedcounterclockwiseabouttheorigin,
Forarotationof90°,(𝑥, 𝑦) ⟶ (−𝑦, 𝑥).
Forarotationof180°,(𝑥, 𝑦) ⟶ (−𝑥, −𝑦). Forarotationof270°,(𝑥, 𝑦) ⟶ (𝑦, −𝑥).
Theorem9.4:CompositionTheorem
Acompositionoftwo(ormore)isometriesisanisometry.
Theorem9.5:ReflectionsinParallelLinesTheorem
Areflectionofafigureover/in/abouttwoparallellinesisATRANSLATIONOF
LENGTHTWICETHEDISTANCEBETWEENTHEPARALLELLINES.
Theorem9.6:ReflectionsinIntersectingLinesTheorem
Areflectionofafigureover/in/abouttwointersectinglinesisAROTATIONOF
DEGREETWICETHEANGLEFORMEDBYTHEINTERSECTINGLINES.
Theorem5.1:MidsegmentTheoremThesegmentconnectingTHEMIDPOINTS
OFTWOSIDESOFATRIANGLE(THEMIDSEGMENT)ISHALFTHELENGTHOFTHE
THIRDSIDEANDPARALLELTOTHETHIRDSIDEOFTHETRIANGLE.
Theorem5.2:PerpendicularBisectorTheoremInaplane,ifapointisonthe
perpendicularbisectorofasegment,thenTHEPOINTISEQUIDISTANTFROMTHE
SEGMENT’SENDPOINTS.
Theorem5.3:ConverseofthePerpendicularBisectorTheorem
Inaplane,ifapointisequidistantfromtheendpointsofasegment,thenITISON
THEPERPENDICULARBISECTOROFTHESEGMENT.
Theorem5.4:ConcurrencyofPerpendicularBisectorsofaTriangleTheorem Theperpendicularbisectorsofatriangleintersectatapoint(circumcenter)
thatisEQUIDISTANTFROMEACHOFTHETRIANGLE’SVERTICES.
Theorem5.5:AngleBisectorTheoremIfapointisonthebisectorofanangle,
thenITISEQUIDISTANTFROMTHETWOSIDESOFTHEANGLE.
Theorem5.6:ConverseoftheAngleBisectorTheoremIfapointisintheinterior
CorollaryofTheorem5.8:ConcurrencyoftheMediansofaTriangleTheorem
Thecentroidofatrianglewithvertices(X1,Y1),(X2,Y2)and(X3,Y3)inthex-y
coordinateplaneis…
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Theorem5.9:ConcurrencyoftheAltitudesofaTriangleTheorem
ThealtitudesofatriangleINTERSECTATAPOINT(ORTHOCENTER).
Theorem5.10
Ifonesideofatriangleislongerthananotherside,thenTHEANGLEOPPOSITE
THELONGERSIDEISLARGERTHANTHEANGLEOPPOSITETHESHORTERSIDE.
Theorem5.11
Ifoneangleofatriangleislargerthananotherangle,thenTHESIDEOPPOSITE
THELARGERANGLEISLONGERTHANTHESIDEOPPOSITETHESMALLERANGLE.
Theorem5.12:TriangleInequalityTheorem
ThesumofanytwosidesofatriangleisGREATERTHANTHELENGTHOFTHE
THIRDSIDE.
Theorem5.13:HingeTheoremIftwosidesofatrianglearecongruenttotwo
sidesofanothertriangleandtheincludedangleofthefirstislargerthanthe
includedangleofthesecond,thenTHIRDSIDEOFTHEFIRSTTRIANGLEIS
ofanangleandisEQUIDISTANTFROMTHETWOSIDESOFTHEANGLE,THENIT
LONGERTHANTHETHIRDSIDEOFTHESECONDTRIANGLE.
ISONTHEANGLEBISECTOROFTHEANGLE.
Theorem5.14:ConverseoftheHingeTheorem
Theorem5.7:ConcurrencyofAngleBisectorsofaTriangleTheorem Iftwosidesofatrianglearecongruenttotwosidesofanothertriangleandthe
Theanglebisectorsofatriangleintersectatapoint(incenter)thatis
thirdsideofthefirstislongerthanthethirdsideofthesecond,thenINCLUDED
EQUIDISTANTTOEACHOFTHESIDESOFTHETRIANGLE.
ANGLEOFTHEFIRSTTRIANGLEISLARGERTHANTHEINCLUDEDANGLEOFTHE
Theorem5.8:ConcurrencyoftheMediansofaTriangleTheoremThemedians
SECONDTRIANGLE.
ofatriangleintersectatapoint(CENTROID)thatdivideseachmedianintoa
vertex-sidepartthatisTWO-THIRDSthemedian’slength,andamidpoint-side
partthatisONE-THIRDthemedian’slength.