Auxiliary Lines - MathBitsNotebook (Geo - CCSS Math)

In geometry, it may be necessary to add a line or segment to a diagram to help in solving a problem or proving a concept. Such an added line or segment is ... AuxiliaryLines MathBitsNotebook.com TopicalOutline|GeometryOutline| MathBits'TeacherResources TermsofUse  ContactPerson:DonnaRoberts Ingeometry,itmaybenecessarytoaddalineorsegmenttoadiagramtohelpinsolvingaproblemorprovingaconcept.Suchanaddedlineorsegmentiscalledanauxiliaryline.Theword"auxiliary"meansprovidingadditionalhelporsupport. Whileitisneveracceptabletochangeanyoftheoriginalpartsofadiagram,itisacceptabletoaddnewlinesorsegments.Thesenewlinesaredrawn"dashed"or"dotted"toindicatedthattheywerenotpartoftheoriginaldiagram. Auxiliarylinesareoftenusedtocreateatriangleinadiagramortocreatetwocongruenttrianglesinadiagram.Yourgoalistoaddsomethingtothediagramthatwillhelpyousolvetheproblem. Thereareaseriesofpostulatesthatareusedtosupporttheadditionofanauxiliarylinetoaproof,suchas: • Twopointsdetermineoneuniquelineorsegment. • Eachanglehasoneuniqueanglebisector. • Throughapointnotonaline(segment),onlyonelinecanbedrawnparalleltothegivenline(segment).(ParallelPostulate) • Throughapointnotonaline(segment),onlyonelinecanbedrawnperpendiculartothegivenline(segment). • Throughapointnotonasegment,onlyoneline(segment)canbedrawntothemidpointofthegivensegment. Example: Theproblemattheleftwantsustoprovethatthebaseanglesofanisoscelestrianglearecongruent.Asyoucansee,thereisnotmuchforustoworkwithinthediagram. Theauxiliarysegmentgivesustwotriangleswithwhichtowork.   IfweaddanauxiliarylinewhichisaperpendiculartothebasefrompointB,wecanformtworighttrianglesinwhichtowork.Wecanthenworkourwaytowardgettingtheanglestobecongruentascorrespondingpartsofcongruenttriangles. Alwaysaddanauxiliarylineinthesimplestwaypossible.Donotaddamultitudeofauxiliarylinesandmakeaconfusingmessofthediagram. Besurethattheconditionsthatyouplaceonanauxiliarylineareactuallytrue. Considerthisexample: Giventhediagramattheleft,astudentdrawsanauxiliarysegmentfrompointDtopointAformingasecondtriangle.Thestudentstatesthatthe"Reason"fortheauxiliarylineisthat"throughapointnotonaline(segment),onlyonelinecanbedrawnperpendiculartothegivenline(segment)"andstatesthat.WRONG! WehavenowayofknowingthattheperpendicularfromAtowillintersectatpointD.ItwillmorelikelyintersectsomewherebetweenpointCandpointD.ThestudentcandrawthesegmentfromAtoD,statingthattwopointsdetermineoneuniqueline,butthestudentcannotstatethatthesegmentisperpendiculartoatD. NOTE:There-postingofmaterials(inpartorwhole)fromthissitetotheInternet iscopyrightviolation andisnotconsidered"fairuse"foreducators.Pleasereadthe"TermsofUse". TopicalOutline|GeometryOutline|MathBitsNotebook.com|MathBits'TeacherResources TermsofUse   ContactPerson:DonnaRoberts