BigIdeasCh6Notes

Geometry Name___________________________________________ Date____________________________Period__________

Chapter6

Ch6.1PerpendicularandAngleBisectors

PerpendicularBisector:______________________________________________________________________

Whatisspecialabouttriangles∆APCand∆BPC?

a)EG= FH=

�

b)x= CD=

�

Whatisthedistancefromapointtoaline?________________________________________________

___________________________________________________________________________________

AngleBisector:________________________________________________________________________________

Perpendicular Bisector Theorem

Converse of Perpendicular Bisector Theorem

Angle Bisector Theorem

�1

BigIdeasCh6Notes

a)� = GF=

�

b>x= PS=

�

Prac8ce:WriteanequaNonoftheperpendicularbisectorofthesegmentwithendpointsP(-2,3)andQ(4,1).

Ch6.2BisectorsofTriangles

Perpendicularbisectorsofatriangleintersectatthe___________________________________________

Circumcenterisequidistantfromall________________________________.

Converse of the Angle Bisector Theorem

∠GFJ

�2

BigIdeasCh6Notes

Loca8onsofCircumcenter

Prac8ce:FindthecoordinateofthecircumcenterofthetrianglewiththeverNces:O(0,-9),Y(0,0),Z(8,0)

Anglebisectorsofatriangleintersectatthe___________________________________________

Incenterisequidistantfromall_____________________________.

Circumcenter Theorem

�3

BigIdeasCh6Notes

Prac8ce:MPandLPareanglebisectorsof∆LMN.Findeachmeasure.

1)thedistancefromPtoMN

2)� =

Ch6.3MediansandAl8tudesofTriangles

Median:___________________________________________________________

Thethreemediansofatrianglemeetatthe_______________________________.

Prac8ce:IfDC=21andXE=4,solveforthefollowinglengths.

CX=

AE=

Incenter Theorem

m∠PMN

Centroid Theorem

�4

BigIdeasCh6Notes

Prac8ce:Findthecoordinatesofthecentroidof∆RSTwithverNcesR(2,1),S(5,8),T(8,3).

Al8tudeofatriangleis____________________________________________________________________________________

ThethreealNtudesmeetatthe____________________________________.

Loca8onsofOrthocenter

Prac8ce:Findthecoordinatesoftheorthocenterof∆XYZwithverNcesX(-5,-1),Y(-2,4),Z(3,-1).

�5

BigIdeasCh6Notes

Thecoincidentpointsyoushouldknowrightnow,andhowtofindeachofthesepoints.

Ch6.4TheTriangleMidsegmentTheorem

TriangleMidsegment:_________________________________________________________________

Determinethevaluesofx,y,andz.

Triangle Midsegment Theorem

�6

BigIdeasCh6Notes

SolveforJL,PM,� .

Exercise:TheverNcesof△RSTareR(-7,0),S(-3,6),andT(9,2).MisthemidpointofRT,

andNisthemidpointofST.ShowthatMN||RSand� .

Ch6.5IndirectProofandInequali8esinOneTriangle

DirectProofflowchart:

IndirectProofor“___________________________________________”

m∠MLK

MN = 12RS

�7

BigIdeasCh6Notes

Given: AnytriangleProve: Atrianglecannothavetwoobtuseangles.

1)IdenNfytheconjecturetobeproven:

2)Assumetheopposite(negaNon)oftheconclusionistrue.

3)UsedirectreasoningtoshowthattheassumpNonleadstoacontradicNon.

4)ConcludethatsincetheassumpNonisfalse,theoriginalconjecturemustbetrue.

Trythis:Ordertheanglesfromsmallesttolargest.Then,usingtheangleorder,canyouorderthesidelengthsfromsmallesttolargest?

Exercise:1)Listtheanglesfromsmallesttolargest

�

2)Listthesidesfromshortesttolongest.

�

Triangle Larger Angle Theorem

Triangle Longer Side Theorem

�8

BigIdeasCh6Notes

Atrianglecanbeformedby3segments,butnoteverysetofthreesegmentswillwork.

Howareyousupposedtoknow?

Canyoumaketrianglesoutofthefollowinglengths?

a)8,12,21

b)6.2,7,9

c)4.3,5.7,10

Exercise:Thefigureshowsapproximatedistances.WhatistherangeofdistancesfromSanFranciscotoOakland?

Triangle Inequality Theorem

�9

BigIdeasCh6Notes

Ch6.6Inequali8esinTwoTriangles

Defini8on:Whentwosidesofatrianglestaythesamelengthandthethirdsidechangeslength,itiscalled______________________________

Iftheincludedangleofthetwosidesgetsbigger,thenthethirdsidegets__________________________________

Example:Whichisgreater?SideJMorML?

Problem:TwogroupsofbikersleavethesamecampheadinginoppositedirecNons.Eachgrouptravels2miles,thenchangesdirecNonandtravels1.2miles.GroupAstartsdueeastandthenturns45°towardnorth.GroupBstartsduewestandthenturns30°towardsouth.Whichgroupisfartherfromcamp?Explainyourreasoning.

Hinge Theorem

Converse of theHinge Theorem

�10