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S)\  `&Times New RomanS 64Hyperlink    %2A`Arial\  `&Times New Romanhttp://www.keypress.com/sketchpad/http://mathforum.org/sketchpad/sketchpad.htmlhttp://www.math.com/school/subject3/lessons/S3U2L3GL.htmlhttp://www.mdabbs.fslife.co.uk/Maths_Pages/SketchPad_Files/Quadrilaterals/Quadrilaterals.htmlhttp://www.mste.uiuc.edu/m2t2/geometry/quadsDiag.html'dxd(3$ !  ݁Level 1Level 2Level 3Level 4Level 5( $ Figure  1  _ZaX I&mage <=8C HKKKK !  _b@*@{{6XX EXPLORINGQUADRILATERALS z @2 2 !ANDPARALLELOGRAMS #XqX6d#Ԉ V  @[[%PREPAREDBY (B$% @//%MIKE_NEDROW_Ԉ )0%& @$$(2001 *&'  -(* Ї@#!X}LXXXq !`X!X}LQuadrilaterals  @ ExploringParallelograms #G!`#Ԉ k #XqXG3#ThisGeometers_Sketchpad_Ԁactivitywillinvestigatequadrilateralsandparallelogramswhicharea t specialtypeofquadrilateral. b  A quadrilateral canbedefinedas4coplanarsegmentswhichintersectonlyattheirendpoints >  (eachendpointbelongstoonlytwosegments). , | BeginbyopeningtheGeometers_Sketchpad_Ԁsoftware.Usingthesegmenttool,constructa X  quadrilateral,similartotheoneshownbelow.Afteryouhaveconstructedit,labelthevertices F  usingthelabeltool. 4   ??(#(# ????qi=-)`~0  `FE  ?|Lq ???? ???? ?????? ? ???? ???? ????   `   (#(#??Quadrilateral_ABCD_Ԁiscalled convex .Aconvexpolygonisonethatnoexternalsegmentcanbe T drawntoconnectanytwoverticesofthepolygon. B Now,measurealloftheinterioranglesforthequadrilateralyouhaveconstructed.Remember,to n measureanangle,youmustselecttheendpointsofthetwosegmentsthatformtheangleandthen  \ gototheMeasuremenuandselectAngle. J Oncealloftheangleshavebeenmeasured,gobacktotheMeasuremenuandselectCalculate. & Now,sumtheanglesforthequadrilateral.     (#(#     qi=-)`~0U!D'  ` / E9 !D' D' }! q                                                                      -(+   Now,compareyourmeasurementstothoseofyourclassmates.  Whatdidyoudiscover?___________________________________________________________  ______________________________________________________________________________  Byselectingoneoftheverticesusingtheselecttool,theshapeofthequadrilateralcanbechanged. h Asyouchangetheshape,thecomputerwillrecalculatetheanglemeasurements.Youshould V  noticethatforanyconvexquadrilateral,thesumoftheangleswillequal360$Concave D  quadrilateralshavethesameproperty,however,Geometers_Sketchpad_Ԁwillnotshowthiswiththe 2  quadrilateralyouconstructedduetothewayitmeasures_angles._  p  Selectanew_Sketchpad_ԀwindowbyselectingFileandthenNewSketch.Wearenowgoingto  L  exploretheworldof parallelograms. Aparallelogramisaquadrilateralwhereoppositesidesare :  parallel.ParallelogramshaveseveralpropertiesthatwecandiscoverusingGeometers (  _Sketchpad_.   First,weneedtoconstructaparallelogram.Itisimportantthatitbeconstructedinaprecise   mannersothatnomatterhowwechangetheshape,itwillremainaparallelogram.  Step1:0 ` Usethesegmenttooltoconstructtwosegmentsconnectedatoneendpoint.~` (#` (# Step2:0 ` Usetheselecttooltoselectonesideandtheendpointoftheothersegment(the Z endpointthatthetwosegmentsdonotshare).GotoConstructandselectParallel H Line.Now,usethesameprocessandselecttheothersegmentandtheendpoint. 6 OnceagainselectConstructandParallelLine.$t` (#` (#  (#(#qi=-)`~0\  `E\ \ G Z q  GGqi=-)`~0@   ` Ec@   5 q GGGG GGGG GGGG GGGG GGGG GGGG GGGG GGGG@  GGGG GGGG GGGGGGG GG (#(#  Step10 0 (# (#0h(#(#0h(#h(#0(#(#0p(#(#Step2'f"%p(#p(#  -(, ЇStep3:0 ` Wherethetwolinesintersect,usethepointtooltoplaceapoint.Next,selectthe  twolinesandgototheDisplaymenuandchooseHideLines.Nowusethesegment  tooltoconnecttheendpointsofthetwosegmentstotheotherpointtoforma  parallelogram.Usethesegmenttooltoconstructthetwodiagonalsandplaceapoint z attheirintersection.Finally,usethelabelfeaturetolabeltheverticesaswellasthe h pointofintersectionofthediagonals.0` (#` (#V (#(# qi#=-)`~0   `fE7   U #  q   (#(#                                                     `  (#(#  Step4:0 ` UsingtheMeasuremenuandtheselectiontool,measurethelengthofeachside,the l measureofeachinteriorangleandthemeasurementsofeachdiagonalZ` (#` (# qi=-)`~01 z `iDEL  z z q   (#(#                                                                                 (#(#  Again,themeasurementsforyourquadrilateralwilllikelyvaryfromthoseinthediagramabove.If 'f"% youexaminethemeasurements,youwillnoticethatsomeofthemeasurementsareequal. (T#& Geometers_Sketchpad_Ԁallowsustochangethedimensions/shapeofthequadrilateralwhilekeeping (B$' theshapeaparallelogram.Tryselectingoneoftheverticesanddraggingyourmousetochangeits )0%( size/shape.Asyoudothisnoticewhathappenstothemeasurements.Whatpropertiesof *&) parallelogramscanyouconjecturefromthisexercise?Usethefollowingpagetowriteyour + '* answer. `     ,'+  -(, ______________________________________________________________________________  ______________________________________________________________________________  ______________________________________________________________________________  ______________________________________________________________________________ z ______________________________________________________________________________ h ______________________________________________________________________________ V  ______________________________________________________________________________ D  ______________________________________________________________________________ 2  ______________________________________________________________________________  p ______________________________________________________________________________  ^  ______________________________________________________________________________  L     `     -(, Thisexerciserevealedseveralpropertiesofparallelograms:   (5h83" 3"     3(25h  1  .3  0 `   Oppositeanglesofaparallelogramarecongruent 3((݌` (#` (# Ќ  " 3"     3)25h  2  .3  0 `   Oppositesidesofaparallelogramarecongruent 3))݌` (#` (# Ќ  " 3"     3*25h  3  .3  0 `   Consecutiveanglesofaparallelogramaresupplementary 3**݌z` (#` (# Ќ  " 3"     3+25h  4  .3  0 `   Thediagonalsofaparallelogrambisecteachother. 3+,݌h` (#` (# Ќ   XXqBRAINSTORMING:(GroupWork) D  #XqX ,# Thelastpartofthisactivitywillinvolveinvestigationofthediagonalsofaparallelogram. _  Constructaparallelogramandcalculatethetotallengthofeachdiagonalaswellastheangles M   formedbytheintersectionofthediagonals.Inaddition,youwillneedtofindthelengthofeach ;  sideaswellasthemeasureoftheinteriorangles. )y     `     h UsethedragfeatureofGeometers_Sketchpad_Ԁtoanalyze(lengthofside/interiorangles) U  parallelogramswhen: C  1.0  Thediagonalsareperpendicular.  ________________________________________________________________________   ________________________________________________________________________  ________________________________________________________________________  ________________________________________________________________________(#(#   ________________________________________________________________________ u   2.0  Thediagonalsarecongruent. -} ________________________________________________________________________ k ________________________________________________________________________  Y ________________________________________________________________________ G ________________________________________________________________________5(#(#   ________________________________________________________________________  # 3.0  Thediagonalsarebothperpendicularandcongruent. $" ________________________________________________________________________ y% # ________________________________________________________________________ g&!$ ________________________________________________________________________ U'"% ________________________________________________________________________C(#&(#(#   ________________________________________________________________________ 1)$'    `    ,9(+ b    ~ XXqRHOMBUS0  0h(#(#Aparallelogramwherethe   diagonalsareperpendicularh(#h(# RECTANGLE0 h Aparallelogramwherethe   diagonalsarecongruentaf h(#h(#    `     h Allinterioranglesequal90$   SQUARE0  0h(#(#Aparallelogramwherethe 6;  diagonalsareperpendicularand   congruentX] h(#h(# 0  0` (#(#0 ` (#` (#0 (# (#0h(#(#Allsidesofasquareare  congruentzh(#h(# 0  0` (#(#0 ` (#` (#0 (# (# h Allinterioranglesequal90$ @  @  @  @  @  (#(# 0  0` (#(#0 ` (#` (#0 (# (#0h(#(#Asquareisbotharhombusand  arectangle#XqX ~7# -!2h(#h(#    `   -( @""$ )XqXXXq)~ X)XqWEBRESOURCES  #)XqX )~X<#)!=X)Xq #!=)!=9<#(̊!= Homeof_Geomter_s_Sketchpad_Ԁ(KeyCurriculumPress) > C 4* O  5  '  http://www.keypress.com/sketchpad/'=  6==  7 8=   &(̊(̊    &  &>  _Geomter_s_Sketchpad_ԀResources&>i?  E?Ԍ   Ќ(̊(̊(̊(̊  4+ O  5  '  http://mathforum.org/sketchpad/sketchpad.html'@  6x@@  7 `@']?    AFirstGlanceatQuadrilaterals   4, O  5  '  http://www.math.com/school/subject3/lessons/S3U2L3GL.html'_B  6B'B  7 UB fk  AdvancedQuadrilateralActivitieswithGeometers_Sketchpad_  NS 4- O  5  '  http://_www.mdabbs.fslife.co.uk_/_Maths_Pages_/_SketchPad_Files_/Quadrilaterals/_Quadrilaterals.html_'1D  6CC  7 'D   AWebSimulationActivitytoDiscoverQuadrilaterals   4. O  5  '  http://www.mste.uiuc.edu/m2t2/geometry/quadsDiag.html'}F  61FEF  7 HsF  #!=(̊=##XqX!=<#