Unit6GuidedNotes Name:__________________________Geometry Period:_____Task:Todiscovertherelationshipbetweenthelengthofthemid-segmentandthelengthofthethirdsideofthetriangle.Materials:Thispaper,compass,rulerSteps:1.Usingastraightedge,constructatriangle.Dothisnexttotheimageofatriangleabove,butdon’tmakeyourtrianglecongruenttoit.2.Usingacompass,bisecteachsideofthetriangletolocatethemidpointofeachside.3.Connectthemidpointstoformthethreemidsegments.4.Measurethemidsegmentsandthethirdsidesforeachpairing.Recordthelengths.5.Recordthedataforyourtriangle.Compareyourresultswithyourgroupandmakeaconjectureregardingtherelationshipbetweenthelengthofthemidsegmentandthelengthofthethirdsideofthetriangle.Data:Conjecture:Wethinkthatamidsegmentandthethirdsideofatriangle….

TheTriangleMidsegmentTheorem:Ifasegmentjoinsthemidpointsoftwosidesofatriangle,then

_______________________________________________________________________ Practice 1.Namethetrianglesidethatisparalleltothegivensegment.a. XY b. 𝑋𝑍 c. ZY 2.PointsM,N,andParethemidpointsofthesidesof∆QRS.QR=30,RS=30,andSQ=18.3.FindMQ. 4.FindMP. 5.FindPN

Midsegment ThirdSideDE= BC=EF= AC=FD= BA=

ProofsGiven:𝑈𝑋 ≅ 𝑋𝑊,𝑊𝑌 = !

!𝑊𝑉

Prove: 𝑋𝑌isamidsegmentof∆𝑈𝑊𝑉.

Given: 𝑋𝑌isamidsegmentof∆𝑈𝑊𝑉.Prove:∠𝑊𝑋𝑌 ≅ ∠𝑋𝑈𝑉

Statements Reasons

Statements Reasons

ThePerpendicularBisectorTheorem:Ifapointisontheperpendicularbisectorofasegment,then

_________________________________________________________________Sketch:

TheConverseofthePerpendicularBisectorTheoremisalsotrue:Ifapointisequidistantfromtheendpointsofasegment,then

_________________________________________________________________

Sketch:PracticeFindx,thenfindIK. Drawtheperpendicularbisectorof𝐴𝐵.ProofsGiven:𝑊𝑋𝑌𝑍 isarhombus.Prove: ∆𝑊𝑉𝑌 ≅ ∆𝑍𝑉𝑋

Statements Reasons

AngleBisectorTheorem:Ifapointisonthebisectorofanangle,then

______________________________________________________________________________________________ Sketch:

TheConverseoftheAngleBisectorTheoremisalsotrue:Ifapointintheinteriorofanangleisequidistantfromthesidesofanangle,then

________________________________________________________________________________________________Sketch:PracticeHowis𝑀𝑅 relatedto𝑃𝑅?Howdoyouknow?Find𝑀𝑅and𝑃𝑅.Findthevalueofthevariable.Then,identifywhichtheoremyouusedtowritetheequation.x=____________ x=___________ Theoremused: Theoremused: x=___________ y=___________

Theoremused: Theoremused:

ProofsGiven:𝑃𝑀 ⊥ 𝑂𝑃,𝑀𝑁 ⊥ 𝑂𝑁,𝑃𝑀 ≅ 𝑀𝑁 𝑚∠𝑃𝑂𝑀 = 𝑥 + 17 °,𝑚∠𝑁𝑂𝑀 = 3𝑥 − 5 °Prove: 𝑥 = 11

PointsofConcurrencyWhenthreeormorelinesintersectatonepoint,theyare_______________________.An________________________ofatriangleisa___________________segmentfromavertexofthetriangletothelinecontainingtheoppositeside.ConcurrencyofAltitudesTheorem:Thelinesthatcontainthe___________________ofatriangleareconcurrent.Thispointofconcurrencyiscalledthe_________________________.Sketchesofaltitudes:Theorthocentercanbeinside,outside,oronthetriangle.

Statements Reasons

A__________________ofatriangleisasegmentwhoseendpointsarea__________andthe_____________________oftheoppositeside.ConcurrencyofMediansTheorem:The________________________________________ofatriangleareconcurrentatapointthatis_______thedistancefromeachvertextothemidpointoftheoppositeside.Thispointofconcurrencyiscalledthe_______________________.PointGisthecentroidof∆ABC,AD=8,AG=10,BE=10,AC=16andCD=18.Findthelengthofeachsegment.

DB=__________ EA=__________CG=__________ BA=__________GE=__________ GD=__________BC=__________ AF=__________

ConcurrencyofPerpendicularBisectorsTheorem:The_________________________________________________________________ofthesidesofatriangleareconcurrentatapointequidistantfromthevertices.Thispointofconcurrencyiscalledthe_________________________.

Thecircumcentercanbeinside,outside,oronthetriangle.

Inthediagram,pointOisthecircumcenter.Findtheindicatedmeasure.MO=___________ PR=__________MN=__________ SP=__________m∠MQO=__________IfOP=2x,findx.

x=__________

ConcurrencyofAngleBisectorsTheorem:The________________________________________oftheanglesofatriangleareconcurrentatapointequidistantfromthesidesofthetriangle.Thispointofconcurrencyiscalledthe_____________.Inanequilateraltriangle,allfourpointsofconcurrencyarethesamepoint!

Practice

IssegmentABamidsegment,perpendicularbisector,anglebisector,median,altitude,ornoneofthese?

B

AA

B

A

B

AB

A

BA

B

Inthediagram,pointGisthecircumcenter.Findtheindicatedmeasure.

AG=__________ BG=__________CF=__________ AB=__________FC=__________ GF=__________m∠ADG=__________IFBG=(2x–15),findx. x=__________

PointTistheincenterofΔPQR.ST=__________IfTU=(2x–1),findx.x=__________Ifm∠PRT=24º,thenm∠QRT=__________Ifm∠RPQ=62º,thenm∠RPT=__________

PointSisthecentroidofΔRTW,RS=4,VW=6,andTV=9.Findthelengthofeachsegment.RV=__________ SU=__________RU=__________ RW=__________TS=__________ SV=__________

PointGisthecentroidof∆ABC.Usethegiveninformationtofindthevalueofthevariable.IfFG=x+8andGA=6x–4,x=__________IfCG=3y+7andCE=6y,y=__________