Postulates and Theorems



 TextbookReference:Pages682‐693

PostulatesandTheorems‐Geometry

Page1

Topic1–ToolsofGeometry

Postulate1‐1:

Throughanytwopoints,thereisexactlyoneline.

Postulate1‐2:

Iftwodistinctlinesintersect,thentheyintersectinexactlyone

point.

Postulate1‐3:

Iftwodistinctplanesintersect,theyintersectinexactlyone

line.

Postulate1‐4:

Throughanythreenoncollinearpointsthereisexactlyone

plane.

RulerPostulate:

Everypointonalinecanbepairedwitharealnumber.This

makesaone‐to‐onecorrespondencebetweenthepointson

thelineandtherealnumbers.

SegmentAdditionPostulate:

IfthreepointsA,BandCarecollinearandBisbetweenAand

thenAB+BC=AC.

ProtractorPostulate:

Consider



andapointonAononesideof



.Everyray

oftheform



canbepairedonetoonewitharealnumber

from0to180.

AngleAdditionPostulate:

IfpointBisintheinteriorof

AOC

,then

m AOB m BOC m AOC



 

.

LinearPairPostulate:

Iftwoanglesformalinearpair,thentheyaresupplementary.



linesrightangles



Topic2–ReasoningandProof

LawofDetachment:

Ifthehypothesisofatrueconditionalistrue,thenthe

conclusionistrue.Insymbolicform:

Ifpqistrueandpistrue,thenqistrue.

LawofSyllogism:

Ifpqistrueandqris

true,thenpristrue.

SymmetricProperty:

If

BCD

,then

CD AB

.

If

B,then

A.

TransitiveProperty:

If

BCD

,and

CD EF

,then

BEF .

If



 ,and





,then

AC

.

If



 ,and





,then

AC

.

ReflexiveProperty:

BAB

and

A.

VerticalAnglesTheorem:

VerticalAnglesarecongruent.

CongruentSupplementsTheorem:

Iftwoanglesaresupplementsofthesameangle(orcongruent

angles),thenthetwoanglesarecongruent.

CongruentComplementsTheorem:

Iftwoanglesarecomplementsofthesameangle(orcongruent

angles),thenthetwoanglesarecongruent.

Allrightanglesarecongruent.

Iftwoanglesarecongruentandsupplementary,theneachisa

rightangle.



Topic3–ParallelandPerpendicularLines

Same‐SideInteriorAnglesPostulate:

Ifatransversalintersectstwoparallellines,thenthesame‐side

interioranglesaresupplementary.

AlternateInteriorAnglesTheorem:

Ifatransversalintersectstwoparallellines,thenalternate

interioranglesarecongruent.

CorrespondingAnglesTheorem:

Ifatransversalintersectstwoparallellines,thecorresponding

anglesarecongruent.

AlternateExteriorAnglesTheorem:

Ifatransversalintersectstwoparallellines,thenalternate

exterioranglesarecongruent.

ConverseoftheCorrespondingAnglesTheorem:

Iftwolinesandatransversalformcorrespondinganglesthat

arecongruent,thenthelinesareparallel.

ConverseoftheAlternateInteriorAnglesTheorem:

Iftwolinesandatransversalformalternateinterioranglesthat

arecongruent,thenthetwolinesareparallel.

ConverseoftheSame‐SideInteriorAnglesPostulate:

Iftwolinesandatransversalformsame‐sideinteriorangles

thataresupplementary,thenthetwolinesareparallel.

ConverseoftheAlternateExteriorAnglesTheorem:

Iftwolinesandatransversalformalternateexteriorangles

thatarecongruent,thenthetwolines

areparallel.





 TextbookReference:Pages682‐693

PostulatesandTheorems‐Geometry

Page2

Iftwolinesareparalleltothesameline,thentheyareparallel

toeachother.

Inaplane,iftwolinesareperpendiculartothesameline,

thentheyareparalleltoeachother.

PerpendicularTransversalTheorem

Inaplane,ifalineisperpendiculartooneoftwoparallellines,

thenitisperpendiculartotheother.

ParallelPostulate

Throughapointnotonaline,thereisoneandonlyoneline

paralleltothegivenline.

TriangleAngle‐SumTheorem

Thesumofthemeasuresoftheanglesofatriangleis180.

PerpendicularPostulate

Throughapointnotonaline,thereisoneandonlyoneline

perpendiculartothegivenline.

TriangleExteriorAngleTheorem

Themeasureofeachexteriorangleofatriangleequalsthesum

ofthemeasuresofitstworemoteinteriorangles.

TriangleExteriorAngleTheorem–Corollary

Themeasureofanexteriorangleofatriangleisgreaterthan

themeasureofeachofitsremoteinterior

angles.

SlopesofParallelLines

Iftwonon‐verticallinesareparallel,thentheirslopesare

equal.

Iftheslopesoftwodistinctnon‐verticallinesareequal,then

thelinesareparallel.

Anytwoverticallinesorhorizontallinesareparallel.

SlopesofPerpendicularLines

Iftwonon‐verticallinesare

perpendicular,thentheproductof

theirslopesis‐1.

Iftheslopesoftwolineshaveaproductof‐1,thenthelinesare

perpendicular.

Anyhorizontallineandverticallineareperpendicular.

SphericalGeometryParallelPostulate

(Insphericalgeometry)Throughapointnotonaline,thereis

noparalleltothegivenline.

SphericalGeometryTriangleAngleSumTheorem

Insphericalgeometry,thesumofthemeasuresoftheanglesof

atriangleisalwaysgreaterthan180.



Definitions

DefinitionofMidpoint

Amidpointofasegmentisthepointthat

dividesthesegmentintotwocongruent

segments.

DefinitionofRightAngle

Arightangleisananglewhosemeasure

if90.

DefinitionofCongruent Angles

Congruentanglesareanglesthathave

thesamemeasure.

DefinitionofSegmentBisector

Asegmentbisectorisaline,segment,ray

orplanethatintersectsasegmentatits

midpoint.

DefinitionofAngleBisector

Ananglebisectorisaraythatdividesan

angleintotwocongruentangles.

DefinitionofCongruentSegments

Congruentsegmentsaresegmentsthat

havethesamelength.



DefinitionofComplementaryAngles

Twoanglesarecomplementaryifthe

sumoftheirmeasuresis90.

DefinitionofSupplementaryAngles

Twoanglesaresupplementaryifthesum

oftheirmeasuresis180.

DefinitionofIsoscelesTriangle

Anisoscelestrianglehasatleasttwo

congruentsides.



Other

OverlappingSegmentTheorem

Iftwocollinearsegmentsadjacenttoacommonsegmentare

congruent,thentheoverlappingsegmentsformedare

congruent.

OverlappingAngleTheorem

Iftwoanglesadjacenttoacommonanglearecongruent,then

theoverlappinganglesformedarecongruent.



 TextbookReference:Pages682‐693

PostulatesandTheorems‐Geometry

Page3



Topic4–CongruentTriangles

ThirdAnglesTheorem

Ifthetwoanglesofonetrianglearecongruenttotwoanglesof

anothertriangle,thenthethirdanglesarecongruent.

Side‐Side‐Side(SSS)Postulate

Ifthreesidesofonetrianglearecongruenttothreesidesof

anothertriangle,thenthe

twotrianglesarecongruent.

Side‐Angle‐Side(SAS)Postulate

Iftwosidesandtheincludedangleofonetriangleare

congruenttotwosidesandtheincludedangleofanother

triangle,thenthetwotrianglesarecongruent.

Angle‐Side‐Angle(ASA)Postulate

Iftwoanglesandtheincludedsideofonetriangleare

congruentto

twoanglesandtheincludedsideofanother

triangle,thenthetwotrianglesarecongruent.



Angle‐Angle‐Side(AAS)Theorem

Iftwoanglesandanon‐includedsideofonetriangleare

congruenttotwoanglesandanon‐includedsideofanother

triangle,thenthetwotrianglesarecongruent.

IsoscelesTriangleTheorem

Ifwosidesofatrianglearecongruent,thentheanglesopposite

those

sidesarecongruent.

 Corollary

 Ifatriangleisequilateral,thenthetriangleisalso

 equiangular.

Theorem4‐5

Ifalinebisectsthevertexangleofanisoscelestriangle,then

thelineisalsotheperpendicularbisectorofthebase.

Hypotenuse‐Leg(HL)Theorem

Ifthehypotenuseandalegofonerighttrianglearecongruent

tothehypotenuseandalegofanotherrighttriangle,

thenthe

trianglesarecongruent.



Topic6–PolygonsandQuadrilaterals



PolygonAngle‐SumTheorem

Thesumofthemeasuresofann‐gonis



2180n  



Corollary

 Themeasureofeachangleofaregularn‐gonis





2 180n





PolygonExteriorAngle‐SumTheorem

Thesumofthemeasuresoftheexterioranglesofapolygon,

oneateachvertex,is360.



Topic6cont’d–PropertiesofParallelograms



ParallelogramOppsides

 

Ifaquadrilateralisaparallelogram,thenitsoppositesidesare

congruent.

Oppsides



Parallelogram

Ifbothpairsofoppositesidesofaquadrilateralarecongruent,

thenitisaparallelogram

ParallelogramConsec.AnglesareSupp.

Ifaquadrilateralisaparallelogram,thenitsconsecutiveangles

aresupplementary.

Consec.AnglesareSupp Parallelogram

Ifallpairsofconsecutiveanglesinaquadrilateralare

supplementary,thenitisaparallelogram.

ParallelogramOpp.Anglesare





Ifaquadrilateralisaparallelogram,thenitsoppositeanglesare

congruent.

Opp.Angles



 Parallelogram

Ifbothpairsofoppositeanglesofaquadrilateralare

congruent,thenitisaparallelogram.

ParallelogramDiags.bisecteachother

Ifaquadrilateralisaparallelogram,thenitsdiagonalsbisect

eachother.

Diags.Bisecteachother Parallelogram

Ifbothdiagonalsofaquadrilateralbisecteachother,thenitis

aparallelogram.

Anglesupp.tobothconsecanglesParallelogram

Ifanangleofaquadrilateralissupplementarytobothofits

consecutiveangles,thenthequadrilateralisaparallelogram.

Onepairoppsides



and||Parallelogram

Ifonepairofoppositesidesofaquadrilateralisbothcongruent

andparallel,thenitisaparallelogram.

Rhombus  diagonals

Ifaparallelogramisarhombus,thenitsdiagonalsare

perpendicular.



diagonalsRhombus

Ifaquadrilateralisaparallelogramwithperpendicular

diagonals,thenitisarhombus.



 TextbookReference:Pages682‐693

PostulatesandTheorems‐Geometry

Page4

RhombusDiagsbisectopp.Angles

Ifaparallelogramisarhombus,theneachdiagonalbisectsa

pairofoppositeangles.

Diags.Bisectopp.Angles Rhombus

Ifaquadrilateralisaparallelogramwithadiagonalthatbisects

apairofoppositeangles,thenitisarhombus.

RectangleDiags.  

Ifaparallelogramisarectangle,thenitsdiagonalsare

congruent.

Diags.



 Rectangle

Ifaquadrilateralisaparallelogramwithcongruentdiagonals,

thenitisarectangle.

Parallelogramw/



 Diags.Square

Ifaquadrilateralisaparallelogramwithperpendicular,

congruentdiagonals,thenitisasquare.

Kite



Diags.

Ifaquadrilateralisakite,thenitsdiagonalsareperpendicular.

Isoscelestrapezoid



BaseAngles

Ifaquadrilateralisanisoscelestrapezoid,theneachpairof

baseanglesiscongruent.

Isoscelestrapezoid



Diags.

Ifaquadrilateralisanisoscelestrapezoid,thenitsdiagonalsare

congruent.

TrapezoidMidsegmentTheorem

Ifaquadrilateralisatrapezoid,then

1. Themidsegmentisparalleltothebases

2. Thelengthofthemidsegmentishalfthesumofthe

lengthsofthebases.



Topic5–RelationshipsWithinTriangles



TriangleMidsegmentTheorem

Ifasegmentjoinsthemidpointsoftwosidesofatriangle,then

thesegmentisparalleltothethirdsideandhalfaslong.

PerpendicularBisectorTheorem

Ifapointisontheperpendicularbisectorofasegment,thenit



isequidistantfromtheendpointsofthesegment.

ConverseofthePerpendicularBisectorTheorem

Ifapointisequidistantfromtheendpointsofasegment,then

itisontheperpendicularbisectorofthesegment.

AngleBisectorTheorem

Ifapointisonthebisectorofanangle,thenthepointis

equidistantfromthesidesoftheangle.

ConverseoftheAngleBisectorTheorem

Ifapointisintheinteriorofanangleandisequidistantfrom

thesidesoftheangle,thenthepointisontheanglebisector.

ConcurrencyofPerpendicularBisectorsTheorem

Theperpendicularbisectorsofthesidesofatriangleare

concurrentata

pointequidistantfromthevertices.

ConcurrencyofAngleBisectorsTheorem

Thebisectorsoftheanglesofatriangleareconcurrentata

pointequidistantfromthesidesofthetriangle.

ConcurrencyofMediansTheorem

Themediansofatriangleareconcurrentatapointthatistwo‐

thirdsthedistancefromeachvertextothemidpoint

ofthe

oppositeside.

ConcurrencyofAltitudesTheorem

Thelinesthatcontainthealtitudesofatriangleareconcurrent.

TriangleInequalityTheorem

Thesumofthelengthsofanytwosidesofatriangleisgreater

thanthelengthofthethirdside.

Twoanglesof



arenot  Largersideisoppositelarger

angle

Twosidesof



arenot



Largerangleisoppositelarger

side

TheHingeTheorem(SASInequalityTheorem)

Iftwosidesofonetrianglearecongruenttotwosidesof

anothertriangleandtheincludedanglesarenotcongruent,

thenthelongerthirdsideisoppositethelargerincludedangle.

ConverseoftheHingeTheorem(SSSInequality)

Iftwosidesofonetriangle

arecongruenttotwosidesof

anothertriangleandthethirdsidesarenotcongruent,thenthe

largerincludedangleisoppositethelargerthirdside.

