00020001010008main.ACT0002020012eActivity Save.EAC010000004009 geometry theorems.EAC main.ACT%(259 L]E E[4GfTf [QVia Animations qby s , Ravinder Kumar% Alcorn State Univ, ]MS\\rkQ@a?.edu &1 Objectivea ofNis eA)ity to demtrsw elementagemo with helpT animations.[[ How to run an *':[)21.2  Expand  the strip  42.g xClick  on Fpop-up3windowU3U  Resize놐,You will see a geomecal ctruc%. No) 4. _ Edit 5  Aoem 6D Go Once(oroN wo op)   A. Theorem on Adjacent Angles": [$I`Sum of measures a+ created by a line P the same side when it intersects=o$r1is 180.\ Caution׆+pMmswitch $s houting capd:nanimsen.N[9DemoΈ h) `s P`   0B Ov r  A #'L' C uxy h   K4@6 5H8p Y 202`CJ sum=h@3m3mT ]`m AFE:o180-B@2{C9"{2{he EFB:   @6  P FH BACe+`7`Z  BCnC @ Ȑ Ho    "qs% vD6Y80uxI3 ĆBECU0 87CȓFYC@    HW+]@2#!5HUo #@6@tP ` CCx+7ߚȒCDzC3Mo 'rX `WG6!$@     $ 4 Z)A)1Y=[  ,+B. Theorem on Vertically Opposite Angles: <SMeasures of v1o1a1, formed by two intersecting lines are equal.2\DemoΈΊҋfVp`ES#IX`3C)Y@ DvKrA  L  C uxy V _`  N`H F@6P)r!DY`gI:A !UBxRR8  i1Na0iYNȐDA@`g'y4D"XCx GVdR9&`10a3`ho0< 5@ !ˈ#`@%6,P)r!D7 Y`g" FA9h J"`%H ``#zazr` Y0 G6!' d$Šh%~ ADu &&S6p ffIt3&B  A ia"& `vcF%^H cBCi)BVqc vC(,@ވ) @# '%bԆ* +4 @C6JxRR8U Y i1 a0iY.`q,XH vCC3AP+!2H@TO-Y. ׈0/9Ȏ2rU$RQ1wvB ~  :  Fs0{1btGTt1 { % +.-)(('&,!5%-%!E0/-=9 [ &%C. Theorem on Corresponding Angles: 6bWhen a transversal int ectspair ofrallel lines, measuresir clal jthus formed are equal.i\DemoΈ1+xA `P!0A` Y5sFb C#  v r   A 'L' C uxy qq2~@! u3 E@6&ViC#" 86>4H UFCgg9h& `xeD DHb 5H vaCaggh`UBbY7w"{D221h۰C 8@ 9H )A@067Ce7q& `U)dp ;C:c"`lBCV4b)7C;!0< ?"ԂUB 67h`A7? B9  "2@)60 pqP;Y#%% H  : _ |Ck kj CcB^2sp3vTG qA.4DH!`E@ 0CamuK F1iCa` R16<{G5@!H 5"&!YU)dp 440 I!H" *H@168CJdH CICQ`z1!%w(?CK%@ HL]ĘoMwJG!Y& `V@23sNH#]UHpTuGh P G6!_ JňO<DCED A@?>=<IJM34679:8 5!M5CMHFB;85QG [  *D. Theorem on Interior Al nate Angles: :gWhen a transversal i6sectspair ofrallel lines, measuresir :papap thus formed are equal.\DemoΈQU2%d `Ct(@`  gt5Sv  v !&r   A 'L' C uxy qqR@! u E@6d"(Rl  wD@>THCDCw0& `Ip YDIU5@ Ve cT5X1C3%vvHBW"`ACXH B@ 6'``4 2He!d8 CYS%@ CWVZ]Ęo[w"wCw (& ` Rrgw\]UViW / cxsi  ]9 !21!teSuD9b`A  :  ^g "kfq9Q" Y et6 _ `J@6BaA4 `7 T! )-H" CICW87d7YFDCa}@ C_^b^ c { $"  Xa)YXC7HP犆S TdŘh`e02f91L!teSuLb`AL  : % g6q9Q" Y e 1Y H4#`'h, H@%@!>YLJ`aXP1I &1WHȆ9ȓ1 CT i `E2%dP ( CBD=    Ɇg@  ĆʢL\ˢLņ̆g #@6bqF! Y yv S(Yx͢`Ϊ<*Ǝ̐\5ّ=5Ȇ†aAÓ[[N^In the following explorations, you are asked to drag a vertex instead of runn@an animA.fnoTA:[dSelectZandlit withr mouse while keepleft-button (for right-hand8lsett ) pressed. s:8Wstill on geometryndowa1. Make sure thate arrow extreme right of"0toolbar points outward. If not, click on  V.[a2{first ic;>lef is selectik, and itiehleverticaltop-up dr down menu]q().[^33. Now click on the vertex you want to select. That is highl$ted with a squarish dot.[[ TK anolrm(:14Y|must firdealreadyzmll8a%a>CutsideO, away from any objЈ %H. eoremG Medians ofTr gle: [@gMedians of a tr gle are concurrent. The distance+tc roid from= vertex is twi'its4G+=midpoint across. [COdividesRm inp ratio 2:1] O  Z To explore:k Seleca and drag it. \?Demo: Έφ"0a)"W `USq0 Y8pwB0 dD`G$DvKr  A 'L'9C uxy  І9  '7@.6 Y"Q& `wphr6 CJ GB= цg `BgwP. K҆H Gfffffg N Ӝ6W3 gt7 FԊFX%1 X 9  5@6W3 gt"`S0dQa AG= ֆg `gAgP.` YKҎ ל4TwAUߦGE²؊H gEbYf%:ٜ3R%$5H[dCdچ  `C@6#rP"-u YҎ ۆK9  K2KG3 gt& `q wBWK GD= g܊H gD %²ݜ1)PdQa)פBG²іsއ|@%߆ ;dBWy'eS4YFT0 چ ؆@  ߆0P8҆k @z6 va aearXw`B~Y3ц7܆䢘`墲0Ԇ紜8ysF"( YgUg(&X֎J4`zn"ʈPnB醌.ȘB, چԆ .@  ц8<"@L,֎^؆|0"𢖈8d<"@|܆0"8d<"@PĘ2HH @u6|2xIXuP Y Hpph )btJ`چ@  *2 @A6H uTgwS !4YsC HQ(q.`oюJ``z"*0v4GV~ YuQuIQGe֎J`z#"24!=܆؆Ԇ߆݆҆ۆن׆ՆӆЏY[ I. Pythagorean Ther -m:[AIf ABC is a right triangle with =90, then AC=AB+ . HQR To explp:c Selectavertex and drag it.[\@Demo:  A+l=uΈ*.Qa)"B `C %faC H8pwX rH$i5D vKr  A 'L'=^ uxy qqz  6@63v) E df AC^2=2@1 ?9  I1@P6W2* `d5H90nACljH fAfH.`E `CC9huAh Y#03u ,^2Q5W%dRh AB^2+Bmmsmm2@6 2 `Fi5H90  AB= \  eQS} Y ie y 8}JF BUYP@Yo^2+@3 93@@BC@ @ ie QV YwI  )@  25  0)^25 oF4@62) `Y5H90 Hz AC^2=@1 | !W2gwW a9s= Y'U$I SgS!J);z"peq!   09= [Wa a%#J. Theorem on Triangle Inequality:3 [<aSu+f the measures any two sid tN is greater7anj+@2Ĉ"2c$5HBCĆLzBd@ q \["u9t Y uhE h7@  "* @6# Cey'4 #f`FJ`z"pH *!5prRYq5aqRfEJ"`%"pJC=j1u[ .K. Theorem on Sides and Angl of a Tria : [ ?In a t , the greaterEA is oppositesi. POZ To explk Select a vertex drag it.[\@Demo: ULargest Ά#D߈ ``eB .D'vKr  A 'L'[C uxy $ 9  8@6 CG`y+`2`c% G AB= %f fe#0Y YD`3FYQ69qYq&B `A`H8pwE'CB놩) (7@!.D)Ho.*j7jej&iV2 C= l  + @'6. 8q'9 DhUYH BxE.`U',F CGPt;-@#Y. o/ 9QHa &s4q Yi(A&0h`10zV2 9O6PT i87pwABC= +39  !5@(6/ i) `U wA JQ AC=n/~4n4nTnIgt5PnBn%5n3nn  Kᒊ6{2{X B= Y %+7,9  51@<6CXT i) `wt5P ^  A= |/10.-)(4 &,'4 шY65432*$A[&eAct020015geometry theorems.EAC010000004084 geometry theorems.EAC main.ACT%(259 L]E E[4 GhTh [S Via Animations   by  ;; Ravinder Kumar% Alcorn State Univ, hMS\ rkQ@a?.edu &B ObjectivelObjective of this eAity to demonstrate some elementary get theorems withHe helpTanimatiC .[[ Howbrun'*':[)21.2  Expand  jip  42.g xClick  onpop-up3windowU3U  Resize놐,Youll see a9Jtrical c_uon. No) 4. _ Edit 5. Click on Animate [ 6$%Go Once # (or the o wo options)F = A. Theoremb X on Adjacentgles: [fSum of measures a* cre d by a lineՊsame side when it intersects=1 is 180.\Cau )p"Mmswitch݇Y$s houting ca,`PLad sen. Demo ) `s P`   0B DWvKr  A 'L'C uxy qq   K4@6 5H8p Y 2` sum=h@3ډ+m3mT ]`m AFE:o180-@2{C9"2@6 T h `e20 5  EFB:   ` i/ P H BAe+`7 BCnC (@ Ȑ BH[ ` i "qs vD6Y80uxI3 zĆBEP ' `0F@76>6YB)Cj@ e  no +@2#!5HUo #ǘ@t1ʓ C xMMDMz3@  # 'rX2 8`D WG6!$R/ x  $ 4 )A)1Y=[ ,+B. Theorem on Vertically Opposite Angles: <SMeasures of v1o1a1, formed by two intersecting lines are equal.\DemoΈΊ fVp `ES#IX 3C) Y@   <vKr ; A 'L'pC uxy qq` `H F@6P)r!DY`gI“ !UBxRR8  i1Na0iYNȐDA@`g'C4D"XCx CGVdR9 10au``  hA0<;5@ D!oˈ#t@~6P)r!D Y`g" FA9h J"%H`#a!r` Y G6!' d$!Šh%~ ADu &&S6p ffIt3&B ~A ia"vcF%'H  B@6i)& Vqc r `(C@ CT&%;#o%bԆ*8+ xRR8 Y i1 a0iY,C3AP!2H@TO-׈Y.׈0Q/V9ȏ2RrU$RQF1wvB^S  :  `+09  1@ 6'btGTt2  1wvB2>I  :  % +{.-)(('&,!%-%!E/-=9[ &%C. Theorem on Corresponding Angles: 6@yWhen a transversal int ectspair ofrallel lines, measuresir clal thus formed are equal.\DemoΈ 1"&&A `P!0A`E Y5sFb C# DbvKr   A 'L'C uxy qq2@! 3 E@6&ViC#" 8640H UFCgg9hxeD%DHb h5H vaCaggh` UBY7"`D@%6,221h&C 8d@ 9H )AaCe7q& `U)dpm:BV4b)C;!0< ?" UB 67@0@  +?A0cV9 E2U6 pqPxY#%j  : _ |Ck kj CcB2sp3vTG qA4D*H!`EDŘ0CamuKUF1iCa` R1Z < CG(5@! .3H> @M6T5"&!_ YU)dp" 440 IH"XHCCJH IQ`z1!%w(?CK%@ ވL]ĘoMwJG!Y& `V@23sNH#]UHpTuGh P G6! N KLHO@  I<PCED A@?>=>,JM34679: 5!M5eMFB;85QG[  *D. Theorem on Interior Al nate Angles: ;}When a transversal i7sectspair ofrallel lines, measuresir :qaqaq thus formed are equal. \DemoΈQ% .2%d@ `Ct(@`M  gt5S v DjvKr   A 'L'C uxy qqR@!  E6d"(Rl  wD@T8HCDCw0IpYDIhU5@ V cT5Y1C3% vvHB W!"`*A@168CXdH CBC``4U 2He!d8CY%@ CVZ]Ęo[w"wC (&` Rrg \]UViW / cxsi_ ]9 !2G1!teSuD9b`AG  :  ^ " @'6.q9Q"9 Y eE t6.`3_B JCBaA4+7 T! HH"IW87d7YFDCa@ Ȏb^ c  $" Xa)YXC7HP*S TdbŘh`70 f91!teSu`b`A   :  gE  @T6[q9Q"f Y er 1Y H4~hB HCWaA+07 T!iH"GB87d2YFDj @ ȎkogP cHl.W&mb\YZVX[nPo H  F@6bu")C" `9eQ ced SpWH Wdo^q!@gUbakj)r??hihi s*_`*`F T-Gc}Ȉ=9n[XWVmlf]?eUL[ *E. Theorem on Exterior Al +nate Angles: }When a transversal in7sectspair ofrallel lines, measuresir eqaqaq thus formed are equal. \DemoΈt&"XE `6T0$ wAT FfweDvKr   A 'L'+C uxy qq+9  q3@x6(2@Svc 91Q  v5  @D6K AfbV Pi) YVECR"6nww BICdC&AT vT)#xHCHCq2& `52fG Cy@Ȏzov {"3T3V Y $P !sETU|BCD(cY 5ah} "` A@6Uq%iG" !ua8px ~C@  |{]Ę}m9H0Drd8& `'P v{u$@!wx""*2P#%G%GBp@'$Pg ,@6 3T3 Y $P# )GU.`38H  AGCX#I+)`)Sx 7;{"QFAv@!70fUvC@ ĘooS 9H0D%F'`wf/ {xHPii'i$H  C@6 @IA ` XU {=5@ wS\T&Ict Y)v\vvHBhXH"XJCH D`uB&42'%E%H#ވ]@ owEY"sY Qd $QYŘ]UiW Y cxsi H  $|-}&@&&&!&{}@~33v6m ߤ&\zy wx݈1!16!gאMDnum[{ )SF. Theorem on Central Angle and the Subtended by Arc at!CircumferenceZ:c [HsHaHc er of a cTis twiceathw:Ns any pointˈP aining partXZ. \DemoΈ(#&9 `D("8D& .40 DuvKr  A 'L'C uax y  '! $5@76 "& `dA`X9D CJ Twice mKB=$@39 t3t(T"F Y0Sti  c "TbrBP v0P5Az3@H"CXBH'Qs%!CB\A`gw@daC\@   H @W6^ 9v6@i Y"BU%e  ay  3!`DCYYۈ0D9ȏ 0""9A`X9DL < $$*2 9" 4@61 &rP" YI9hT'7.`2  c #cb 4Hw4 YUS'Y  EHz37 A*Y @ YH"o@x $ 8#UhfUDdU19(Bu`0 0@6 "" `9A`X9D     $ 5H  -YP QdEd2J%@!`azRcx4T x xshBX a92kT ik gt5kcirc: q   9  )1@0675H8pB Y gt5P2Bcenter: c 0@@Ę 6 )O0$LIY[M (,G. Theorem on Angle Subtended by DiameE< FnThe Angle subtended by tdiameter at any Point ofcircumference (excep't Cp) s) is a righ@c . \ DemoΈ# / `g2g   DkvKr  A 'L'C uxy qq9  2@6R(/ Y0"f5 c ccb69T Bx Y$Hcww#',  5B@%@!>YLJXaXP1I V &1WHȉȓ1CT i`E2%dP   CBD=    Ɇ\@  ĆʢL\ˢLņ̆g #@6bqF! Y yv S(Yx͢`Ϊ<*Ǝ̐\5ّ=5Ȇ†aAÓ[[N^In the following explorations, you are asked to drag a vertex instead of runn@an animA.fnoTA:[uSelectZandlit withr mouse while keepleft-button (for right-hand8sett ) pressed. s:0W|still on geometryndowVk1. Make sure thate arrow extreme right of"(toolbar points outward. If not, click onO .[2tfirst ic47ylefx is selectid, and ithleverticaltop-up ;dr down menu\p().[^3. NEw ex you wanlo. T|  with a squarish dotf[!To select another vertex:[ 4You must firde63 alreadyIedC.<ll8a%a>Click outsideWO , away fromy obj.[Њ %H. Theorem on Medians of a Tr gle: [@(t( are concurrent.Y distanceS9 croidH is twi'its4Ƈo midpoin cross. [CHdiv s a m in the ratio 2:1]   To explore: Select a vertex and drag it. A\?Demo: CentroidΈφ"in0za)"W `USq0 Y8pwB0 dD`G$DvKr  A 'L'C uxy qqЇ'9  07@76 Y"Qwphr6f GB= цg `gBgwP@ ҆%H  .G*6<fffffgG N Dӆl9 G6GW3 gt&`7 GF= _ԆgȒF%1 ՜5S0dQaAG²֊ `2A`YҎ ׆9  !4@(6/T gt&`wAU2J GE= g؆gH gEgbPYf%:gٜ3R%$5H[CG²ڊ `2CruZYdۜ2dG3dq wBdGD= ܆H  !D@(6P6 %AYKҎ ݆K9 K1K) gt&`PdQa)pBG²іsކ@s߆ dBW'eS4FT0چ ؆``z0 ⪘㴘 vakaeaYrXw`B ц܆@ 0P҆<Ԇo @~68ysF" YgUg (&X֎J袜`z֊"ʈ߆醌.ȘB,ڎr0"J8d<"@||Ј^؆ֆ.@  ц؎"2,Zچ܆b0"|8d<"@Ę2HH @62xIXuP Y Hpph )btJ``z3"* uTgwr !4YsC HQ(q.`zJ`z @  ц" @1680v4GVC YuQuI QGe.`_֎J``z"2چ!܆؆Ԇ߆݆҆ۆن׆ՆӆЎY[  I. Pythagorean Ther m:*3$AIf ABC is a right triangle with =90, then AC=AB+ .j{ To explp:c Selectavertex and drag it.[\Demo:   A+B=CΈ $* ;Qa)"B `C %faC H8pwX rH$ii DwvKr  A 'L'^ uxy ߈q  K6@63v)E df AC^2=@1ۉU9n1n22d5H90nACl  H A@6H `E C `CCC9huAhe Y#03u DG^2oQ 52W%dR AB^2+BC^2=q@s9s2sFi5H90ABl7f lQS) ie y 8p B@6U YP 3@   X^2+e@3; 9Q32* `5H90 BC= f f ie QV YwI ` 0z^2e 4Y AC^2=@1 ^2$ 8!@  A  W2g''Wu @6Fa9s Y'U$I SgSJ)``z܈"p  FR`Ԍ  B [; %#J. Theorem on Triangle Inequality:3 [<aSum of the measures any two sid triangle is greater7an H?In a t , the greaterEA is oppositesi.  To explk Select a vertex drag it.[ \ Demo:  Largest Angle SideΆ6#CINRQ ``eB .DvKr  A 'L'C uxy q $9  8@6%CG`3`2c% AB=\%f fe#0Y YD`3FeYQ69qYq&B `sA6`H8pw `E ', B@k{vafpLG# X[Tݻ&&Tl;ݓOgͫ#u{vS#S،1o߮aDvcΓ'r c*Kwqn־'?[XY*v)Jתf{ u+aҺN%ZLI_7,?6I\P3Α\Zbg%A˗;EWcf2ץnby3خNֱn{WztL׭bq]:֩czVzulu:MǮױ:vMֱXVnԱ9IǞ1Kǖtf;c7uV;c:}VбtG:9XB.FĮnױM:ױY:Yǒ:֧cwX؀Mؠ=cC:6c:Vֱ-:vLulQv۩c/.{YǾcֱt{:[~c{t۫c?ױ}:Mku~~۠c:б)ֱGtlhvؘ=c;ulZv똭c:vPǦt씎}I:ֱXA~cGuM[бt씎H؏u%i:zl7Y.D=l靦5̈R.Azh ޹ӏz1~̋iۋA/ ?o~c]tv=դצcFDOM}lk\/Էr=|ӱ]zFr~Z[n}C9PƲȧerYXVkz|ze߈_lynz=foΦ xƝG4ϩخ}{9'ν*˔'K$=_ɥ[~w)W.ޡffܘWbjNk+Եѡf:^bCD[cU Jm)ΥփvDj:qކ suZs&ܵmO6%k@6 'k[C oL&|Y x$d⩁Oy_د2;k磉R\ͤ[j*=$ZڭڛD0?M^*C-tdY!ov/FkI5ŦY'Z#YK&kDh>I'XDvo6Hv&kdm*YtdmN6 JKΓdJҺ`iMj---,TfBe!_TdtWM :gP}l_H,v/XWUɅUKA־$:?%)fl|eI ~#R"@a!n_ߞ^\ǷUM3_FbVs LȺ۝ _L17,,ߘTK1e"{v*nni1R[7Y:uHGV.}/R/ǗߎԮ>2էvTT^ʆ,e,RzY.r,e]V>#햜ݖ$N.n<$rۑ1YeB{BVn=~XVzOtgO&#eYH1YɩH$NDƓG#/'"/ʦ/FWڃjsjGMZklCYdyG~Y~.ˀ,24ʐԧ'&+R E#_:o, .Ҋ=%S*Գ[VvIz$+_ e>POAj+wDz{B= W*sJVIzʄT%Y_* .+_yE=zެSյfM|,V7G;[=1$MkYlvjiьW_ތ?wON&MT~*gONCjڼжxsS]س3g~Õqw$h'Ou=E}=cXc㢬X[ꫪ͑N{8bvB|t1d]4]arB7ޘ[5ūZӅ #ֲo?/|lCj\m7eWO|u9i$f/d,ι3\C/X,[^ԫW6WjbeV'GϮ]l.G2ϟ7U mM򾷜^{ [¿xU n5{7-}X53R~&s':۽AHdbxglZZ$cS]=1NwF Nu6ɰSj(|zzlRk8w6[q$zҙi3o7 -bL׽39\4cZoN֟γL>4<7։Jg6#{'٨?cgrNyĹ3*G3Q^'?> }69tHMlPw4)iհd<2}$W/㺨x8;[W+# x<[UѽYIY^*'ݔb7~rzOQ/?dP_}qhTTM~.#OwJl9ZK%kkdbW\|ڙ*lm4K:]v@N{1 Bױ߿ԥUbr^g;QѲ~eM83[,`ƶFrYKoWYu-?>{)ZUd7걬l*_Ya\氟Q$iyDLF2.>2n9JWUt/;rcw阕?deTik+s8S$Q[iFsrQ$`9YgɕeTm=.RH}]]~y)jɺXMB5ol b0n`7P:oוϹ, +nw5ήt٤frP"-Z%DZlidRvֲRn+VR}^9pdj3e]5[t5w^MԺ7oQP |ѭr^~ڑ/dc7֌YA-a}[l$e;%] x)y[d7-ugT):%-3NZ]^˗lELڱ89+S2S,IG,^) }AϹjԮwuI Zkz-teyk_/eݼoJNi(`6Y2t3ݢJrim~_ *ok굹j_)kd`ّGw\LYUf!5t;S cQ7ǭm$Ϭ{ɖd>wk%kjO?mXZ#3Hdc{$9nκ}ևDC\2r2gAuluUNŝ{Y g.jK­Wij*f){QZbzljgGgҚ'v )z,m;3'=JVvZL9j?X xeRr=O՛M>֥ Ek6_)wFZ )7@P`J,->z_)[FKԥrR`%< +ճU<>k:}ݭ( ?̕W8O`9^'KR>b}֮C5)N: 'cp3s+Hwjn7L1J?^݇ 54ב$gotQv[I]K]m=>kIW?sF~Qw%>4oL7`>6F}B}XC#7? ;`;p&iؾq1Uk2YX+/UC|zE }[-ޘ4C͍ϤI/58eWqWj&7F,j,&fCC5 '}wjR+_=sD,s%][ks1d.xx♋g.x♋g.x♋g.O\w W. tpﮈ/nbʭezzzPC;PC1bLŘ꓏666kJ* cKܗ/sGn+`9sW3O\\ H_R"1{ %QW3qe 30f` }F,? ? ? ? ? 6{7y}M7y]LJOB0418