andproperties
This is a partiallisting of the morepopulartheorems,postulates
of theseitems.
neededwhenworkingwith Euclideanproofs. You needto havea thoroughunderstanding
Your textb-bok{alrd y.ourteacher)may want y,outorrememberthesetheoremswith slightly different
wording.
Be sureto follow the directionsfrom your teacher.
Properties:
ReflexivePropertyof = or =
SymmetricProperty= or =
Iransitive PropertY= or =
AdditionPropertyof :
SubtractionPropertyof MultiplicationPropertyof :
DivisionPropertyof SubstitutionPropertyof DistributiveProperty
A quantityis congnrent(equal)to itself. a: a
f a : b , t h e nb : a .
lfa:bandb:c.then a-c.
tf equal quantitiesare addedto equal quantities,the sumsare
squal.
tf equalquantitiesaresubtractedfrom equalquantities,the
areequal.
Jifferences
tf equalquantitiesaremultipliedby equalquantities,the
productsareequal. (alsoDoublesof equalquantitiesare
squal.)
tf equalquantitiesaredividedby equalnonzeroquantities,the
luotientsareequal.(alsoHalvesof equalquantitiesareequal.)
for its equalin anyexpression.
A quantitymaybe substituted
f a(b * c), then ab * ac
Angles:
RightAngle Definition
StraightAngle Definition
Right anglesTheorem
AngleAddition Postulate
is 90o.
tf anangleis a rightangle,thenitsmeasure
L
;ftE
is 180".
"ej]'tcg;
lf an angleis a straightangle,then its measure
zfl
All right anglesarecongruent.
h-
ETJ
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f two anglesare adjacent,then their sum is equalto the angle
qon-sharedrays.
brmed
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by
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tf two lines are perpendicular,then they intersectto form a
Perpendicular
Definition
LinearPair Definition
:ightangle.
-+_-
*
Qt'
[f two anglesform a linear pair, then they are adjacentand
heir non-sharedsidesform a straightangle.
f two anglesform a linear pair, then their sum is 1 8 00 .
LinearPairTheorem
VerticalAnslesTheorem
BisectorDefinition
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f two angles ur{ical, thentheyareequal/congruent.
ff
.:) p"\LI r,nLL af LIE LL
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f an angle is bisected,then the bisector splits the angle into
v,-,)congruent/equal
" ^ ' r y ? " - parts.
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Segments:
MidpointDefinition
[f a point is a midpoint, then it splits a segmentinto two
;ongruentparts.
4
4rn;el$o.nL'
ffi#ffi
a_trt_".-+
is bisected,
thenthebisectorsplitsthe segment
f a segment
BisectorDefinition
nrotwocongruenvequal
n"ttrfrt'l
r.c{rrr+ ff;
tf a point is befweentwo endpointsof a segment,then the sum
rf both segmentscreatedis equalto the length of the whole
SegmentAddition Postulate
i
e
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g} Agt g;" AL
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Parallels:
lf two parallel lines are cut by a transversal,then the pairs of
AnglesPostulate
Corresponding
AnglesConverse
Oorresponding
Postulate
)ooes'o"ot'ffi'"5u*t
Lty t-L'-'
*
andthe corresponding
lf two linesarecut by a transversal
linesareparallel.
mglesareconFruent,fhe
--f f
*7p-
f two parallel linei are cut by a transversal,then the alternate
AlternateInteriorAnglesTheorem
AlternateExteriorAngles Theorem
nterior"ffit*S
ffi
f turo parallel lines are cut by a transversal,then the alternate
lxterior anglesare congruent.
-7-
-)
-/
---is*-
SameSideInteriorAnglesTheorem
AlternateInteriorAngles
ConverseTheorem
tf two parallel lines are cut by a transversal,tlie interior angles
rn the sameside of the transversalarersupplementary.
-*/- \
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Lt{tL
-*ryfd,**"__".*.afe $*00!fmfr*t
[f two lines are cut by a transveisaland the alternateinterior
mgles are c/ongrutnt,the lines are parallel.
ff)
il
f two liires are cut by a transversaland the alternateexterior
AlternateExteriorAngles
ConverseTheorem
4
lf two lines are cut by a transversaland the interior angleson
lhe sameside of the transversalare supplementary,the lines
SameSideInteriorAnglesConverse areptrrallel.* I
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tt
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ct
tcr'.enfur1 ft
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Triangles:
(SSS) Congruence
Side-Side-Side
Postulate
tf three sidesof one triangle are congruentto three sidesof
anothertriangle,theq the triangle5
'*- arg congruent.
.t-
*-%--
f two sidesand the included angle of one triangle are
(SAS) Congruence ongruentto the colresponding parts of anothertriangle, the
Side-Angle-Side
Postulate
rianglesarecongruent.
E
fJL=
f two anglesand the included side of one triangle are
(ASA) Congruencerongruentto the coffespondingparts of anothertriangle, the
Angle-Side-Angle
Postulate
rianglesarecongruent. A
4-s
(
f two anglesand the non-includedside of one triangle are
(AAS) Congruence:ongruentto the coffespondingparts of anothertriangle, the
Angle-Angle-Side
Iheorem
(ltH L)
Right-Hypotenuse-Leg
CongruenceThm
riangles
arecongruent.
, ,4
E
f the hypotenuseand leg of one right triangle are congruentto
he correspondingparts of anotherright triangle, the two right
rianglesarecongruent.
q
h_
CPCTC
Iriangle SumTheorem
RemoteInterior Angle Theorem
oresponding parts of congruenttrianglesare congruent.
sum of thejnterior anglesof a triangleis 180'.
^a;?:=-Jc
measureof an exterior angle of a triangle is equalto the
of the measuresof the two remoteinterior angles.
:;)
lsoscelesTriangle Definition
TriangleTheorem
lsosceles
+ rnaArirtgi rnL(--i80
rr'L4?r^Ll 1'nLL
lf a triangleis isosceles,then it haspt leasttwgt^congruent
;ides. H5otrclCS+
A
of- tJ
f two sidesof a triangleare cQngruent,
lhe anglesopposite
hesesidesarecongru"ent.A('+
A
TriangleTheoremConverse f two anglesof a triangle are cqngruent,tbe sidesopposite
lsosceles
heseanglesare congruent. ,{I +
A