,,m>ks' ,,fir/ ,,lessons ,,9 ,,geometry4 ,,9 ,,two ,,p>ts4 ,,objectively ,,pres5t$1 ,,& ,,design$ ,,= ,,! ,,use ,,( ,,prim>y ,,classes ,,9 ,,gramm> ,,s*ools1 ,,academies1 ,,etc4 ,,by ,,b]nh>d ,,m>ks1 ,,pr9cipal ,,( ,,l9coln ,,s*ool1 ,,san ,,francisco4 ,,new ,,york3 ,,publi%$ ,,by ,,ivison1 ,,ph9ney1 ,,blakeman1 ,,co4 ,,philadelphia3 ,j4 ,b4 ,,lipp9cott ,,co4 ,,*icago3 ,s4,c4 ,,griggs ,,co4 #ahfi4 ,ent]$1 accord+ to ,act ( ,congress1 9 ! ye> #ahfh1 by ,,b]nh>d ,,m>ks1 ,9 ! ,cl]k's ,(fice ( ! ,di/rict ,c\rt ( ! ,unit$ ,/ates = ! ,di/rict ( ,cali=nia4 ,geo4 ,c4 ,r& ,av]y1 ,electrotyp]s & ,pr9t]s1 #c ,cornhill1 ,bo/on4 ---------------------------------------- -------------------------------- ,,preface4 ,h[ x ev] came to pass t ,ari?metic %\ld be taught to ! ext5t atta9$ 9 ! gramm> s*ools ( ! civiliz$ _w1 :ile ,geometry is almo/ :olly exclud$ f !m1 is a problem = :i* ! au?or ( ?is little book has (t5 s\ght a solution1 b ) only ?is result2 viz41 t ,ari?metic1 be+ consid]$ an elem5t>y bran*1 is includ$ 9 all sy/ems ( elem5t>y in/ruction2 b ,geometry1 be+ reg>d$ z a high] bran*1 is res]v$ = sy/ems ( advanc$ education1 & is1 on t acc\nt1 rea*$ by b v few ( ! _m :o ne$ x4 ,! error "h is fundam5tal4 ,in/ead ( tea*+ ! elem5ts ( all bran*es1 we tea* elem5t>y bran*es mu* too exhau/ively4 ,! elem5ts ( ,geometry are mu* easi] to le>n1 & are ( m value :5 le>n$1 ?an advanc$ ,ari?metic2 &1 if a boy is to leave s*ool ) m]ely a gramm>-s*ool education1 he w\ld be bett] prep>$ = ! active duties ( life ) a little ,ari?metic & "s ,geometry1 ?an ) m ,ari?metic & no ,geometry4 ,?\s&s ( boys are all[$ to leave s*ool at ! age ( f\rte5 or sixte5 ye>s1 & are s5t into ! c>p5t]-%op1 ! ma*9e-%op1 ! mill-wright's1 or ! surveyor's (fice1 /uff$ to repletion ) ,int]e/ & ,disc\nt1 b s utt]ly ignorant ( ! m]e/ elem5ts ( ,geometry1 t !y c\ld n f9d ! c5tre ( a circle already describ$1 if _! lives dep5d$ ^u x4 ,un?9k+ p]sons frequ5tly ass]t t "y *ildr5 are incapable ( reason+1 & t ! tru?s ( ,geometry are too ab/ract 9 _! nature to be appreh5d$ by !m4 ,to ^! objections1 x may be answ]$1 t any ord9>y *ild1 five ye>s ( age1 c d$uce ! conclusion ( a syllogism if x und]/&s ! t]ms conta9$ 9 ! propositions2 & t no?+ c be m palpable to ! m9d ( a *ild ?an =ms1 magnitudes1 & directions4 ,!re are _m tea*]s :o imag9e t ! p]ceptive faculties ( *ildr5 %\ld be cultivat$ exclusively 9 e>ly y\?1 & t ! reason %\ld be address$ only at a lat] p]iod4 ,x is c]ta9ly true t p]ception %\ld receive a l>g] %>e ( att5tion ?an ! o!r faculties dur+ ! fir/ s*ool ye>s2 b x is equally c]ta9 t no faculty c be safely disreg>d$1 ev5 = a "t4 ,! root does n atta9 maturity be=e ! /em appe>s2 nei!r does ! /em atta9 its gr[? be=e its bran*es come =? to give bir? 9 turn to leaves2 b root1 /em1 & leaves are f\nd simultane\sly 9 ! y\nge/ plant4 ,t ! reason may be pr(itably address$ ?r\gh ! m$ium ( ,geometry at z e>ly an age z sev5 ye>s is ass]t$ by no less an au?ority ?an ,presid5t ,hill ( ,h>v>d ,college1 :o says1 9 ! preface to 8 admirable little ,geometry1 t a *ild sev5 ye>s old may be taught ,geometry m easily ?an "o ( fifte54 ,! au?or holds t ?is sci5ce %\ld be taught 9 all prim>y & gramm> s*ools1 = ! same reasons t apply to all o!r bran*es4 ,"o ( ^! reasons w be /at$ "h1 because x is n suffici5tly recogniz$ ev5 by tea*]s4 ,x is ?is3-- ,! prime object ( s*ool in/ruction is to place 9 ! h&s ( ! pupil ! means ( cont9u+ 8 /udies )\t aid aft] he leaves s*ool4 ,! man :o is n a /ud5t ( "s "p ( ,god's works _c be said to live a rational life4 ,x is ! prop] bus9ess ( ! s*ool to d = ea* bran* ( sci5ce exactly :at is done = read+4 ,*ildr5 are taught to read1 n = ! sake ( :at is conta9$ 9 _! read]s1 b t !y may be able to read all ?r\gh life1 & !reby fulfil "o ( ! requirem5ts ( civiliz$ society4 ,s1 en\gh ( ea* bran* ( sci5ce %\ld be taught to enable ! pupil to pursue x aft] leav+ s*ool4 ,if ?is view is correct1 x is wrong to all[ a pupil to rea* ! age ( f\rte5 ye>s )\t kn[+ ev5 ! alphabet ( ,geometry4 ,he %\ld be taught at lea/ h[ to read x4 ,x c]ta9ly does seem probable1 t if ! y\? :o n[ leave s*ool ) s mu* ,ari?metic1 & no ,geometry1 7 taught ! fir/ rudim5ts ( ! sci5ce1 ?\s&s ( !m w\ld be l$ to ! /udy ( ! high] ma!matics 9 _! mature ye>s1 by reason ( ^? attractions ( ,geometry :i* ,ari?metic does n possess4 ,,to ,,! ,,pr(essional ,,read]4 ,?is little book is con/ruct$ = ! purpose ( in/ruct+ l>ge classes1 & ) ref]5ce to be+ us$ also by tea*]s :o h !mselves no k ( ,geometry4 ,! fir/ /atem5t w acc\nt = ! _m1 & p]haps seem+ly ne$less1 repetitions2 & ! second1 = ! sugge/ive /yle ( "s ( ! que/ions 9 ! lessons :i* develop ! matt] conta9$ 9 ! review-lessons4 ,att5tion is respectfully direct$ to ! foll[+ po9ts3-- ,fir/ ! p>ticul>1 !n ! g5]al4 ,see page #be4 ,:y is m n g an acute angle8 ,:at is an acute angle8 ,"h ! att5tion is direct$ fir/ to ?is p>ticul> angle2 !n ?is is tak5 z an example ( its k9d1 & ! idea g5]aliz$ by describ+ ! class4 ,see also page #bi4 ,:y are ! l9es e f & g h said to be p>allel8 ,:5 are l9es said to be p>allel8 ,_m ( ! que/ions are int5d$ to te/ ! vividness ( ! pupil's conception4 ,see page #bi4 ,also page #gh4 ,if ! circumf]5ce 7 divid$ into #cfj equal p>ts1 w\ld ea* arc be l>ge or small8 ,_m ( ! que/ions are int5d$ to te/ ! att5tion ( ! pupil4 ,! ?+ is n to be recogniz$ by ! def9ition2 b ! def9ition is to be a description ( ! ?+1 a description ( ! conception br\ght to ! m9d ( ! pupil by means ( ! "n4 ,,cont5ts4 ,,"p ,i4 ,,l9es #i ,,po9ts #i ,,crook$ ,,l9es #aj ,,curv$ ,,l9es #aa ,,/raight ,,l9es #aa ,,o!r ,,l9es #aa ,,positions ,,( ,,l9es #ad ,,angles #ag ,,relations ,,( ,,angles #bj ,,adjac5t ,,angles #bj ,,v]tical ,,angles #ba ,,k9ds ,,( ,,angles #bc ,,"r ,,angles #bc ,,acute ,,angles #bd ,,obtuse ,,angles #bd ,,relations ,,( ,,l9es #bg ,,p]p5dicul> ,,l9es #bg ,,p>allel ,,l9es #bh ,,oblique ,,l9es #bh ,,int]ior ,,angles #cj ,,ext]ior ,,angles #ca ,,opposite ,,angles #cb ,,alt]nate ,,angles #cc ,,problems ,,relat+ ,,to ,,angles #ch ,,polygons #dj ,,triangles #dd ,,isosceles ,,triangles #dh ,,problems ,,relat+ ,,to ,,triangles #ec ,,quadrilat]als #ee ,,p>allelograms #ei ,,comp>ison ,,& ,,contra/ ,,( ,,figures #fb ,,measurem5t ,,( ,,surfaces #ff ,,problems ,,relat+ ,,to ,,surfaces #ga ,,! ,,circle ,,& ,,its ,,l9es #gc ,,arcs ,,& ,,degrees #gh ,,p>ts ,,( ,,! ,,circle #hb ,,"p ,,ii4 ,,axioms ,,& ,,!orems4 ,,axioms4 ,,illu/rat$ #he ,,!orems4 ,,illu/rat$ #hh ,,fir/ ,,lessons ,,9 ,,geometry4 ,,"p ,,fir/4 ,,lesson ,,fir/4 ,,l9es4 ,,note ,,to ,,! ,,tea*]4--,9 all ! developm5t-lessons1 ! pupils are to be occupi$ ) ! diagrams1 & n ) ! pr9t$ matt]4 ,see ,note ,a1 ,app5dix4 ,ref] to ,,diagram #a1 & %[ t ,:at are "h drawn are int5d$ to repres5t l5g? only4 ,!y h a little wid?1 t !y may be se54 ,!y are call$ l9es4 ,a l9e is t :i* has l5g? only4 ,,po9ts ,%[ t ,position is d5ot$ by a po9t4 ,x occupies no space4 ,x has "s size1 t x may be se54 ,! ends ( a l9e are po9ts4 ,a l9e may be reg>d$ z a succession ( po9ts4 ,! int]section ( two l9es is a po9t4 ,a po9t is nam$ by plac+ a lett] ne> x4 ,illu/ration3 ,diagram #a4 ,a po9t may be repres5t$ by a dot4 ,! po9t is 9 ! c5t] ( ! dot4 ,a po9t is t :i* d5otes position only4 ,a l9e is nam$ by nam+ ! po9ts at its ends4 ,read all ! l9es 9 ,diagram #a4 ,,crook$ ,,l9es4 ,see ,note ,b1 ,app5dix4 ,does ! l9e m n *ange direction at ! po9t #a8 ,at :at o!r po9ts does x *ange direction8 ,x is call$ a crook$ l9e4 ,a crook$ l9e is "o t *anges direction at "s ( its po9ts4 ,,curv$ ,,l9es4 ,! l9e o p *anges direction at e po9t4 ,x is call$ a curv$ l9e4 ,a curv$ l9e is "o t *anges direction at e po9t4 ,,/raight ,,l9es4 ,does ! l9e i j *ange direction at any po9t8 ,x is call$ a /raight l9e4 ,a /raight l9e is "o t does n *ange direction at any po9t4 ,,o!r ,,l9es4 ,! l9e q r w9ds ab\t a l9e4 ,x is call$ a spiral l9e4 ,! l9e w x w9ds ab\t a po9t4 ,x also is call$ a spiral l9e4 ,a spiral l9e is "o t w9ds ab\t a l9e or po9t4 ,! l9e #g #ha looks l waves4 ,footnote #a3 ,to be read sev51 eight1 n sev5ty-eight4 ,x is call$ a wave l9e4 99 99 99 99 99 ,:at k9d ( a l9e is a b8 ,:y8 ,:at is a /raight l9e8 ,:at k9d ( a l9e is #aa #af8 ,:y8 ,:at is a crook$ l9e8 ,:at k9d ( a l9e is o p8 ,:y8 ,:at is a curv$ l9e8 ,:at k9d ( a l9e is s t8 ,:y8 ,:at k9d ( a l9e is #i #aj8 ,:y8 ,:at is a spiral l9e8 ,:at k9d ( a l9e is w x8 ,:y8 ,,lesson ,,second4 ,,review4 ,read all ! /raight l9es4 7,,diagram #b47 ,:y is m n a /raight l9e8 ,def9e a /raight l9e4 ,read all ! crook$ l9es4 ,:y is #g #h a crook$ l9e8 ,def9e a crook$ l9e4 ,read all ! curv$ l9es4 ,:y is #e #f a curv$ l9e8 ,:at is a curv$ l9e8 ,read all ! wave l9es4 ,read all ! spiral l9es4 ,:y is #c #d a spiral l9e8 ,:y is u v a spiral l9e8 ,:at is a spiral l9e8 ,illu/ration3 ,diagram #b4 ,illu/ration3 ,diagram #c4 ,,lesson ,,?ird4 ,,positions ,,( ,,l9es4 ,let ! pupils hold _! books s t !y w be /raight up & d[n l ! wall4 ,,v]tical ,,l9es4 ,! /raight l9e a b po9ts to ! c5t] ( ! e>?4 7,,diagram #c47 ,x is call$ a v]tical l9e4 ,"n all ! v]tical l9es4 ,a v]tical l9e is a /raight l9e t po9ts to ! c5t] ( ! e>?4 ,,horizontal ,,l9es4 ,! /raight l9e o p po9ts to ! horizon4 ,x is call$ a horizontal l9e4 ,read all ! horizontal l9es4 ,a horizontal l9e is a /raight l9e t po9ts to ! horizon4 ,,oblique ,,l9es4 ,! l9e s t po9ts nei!r to ! c5t] ( ! e>? nor to ! horizon4 ,x is call$ an oblique l9e4 ,read all ! oblique l9es4 ,an oblique l9e is a /raight l9e t po9ts nei!r to ! horizon nor to ! c5t] ( ! e>?4 ,,note4--,aft] go+ ?r\gh ) ! lessons on angles1 ! pupils may be told t oblique l9es are s call$ because !y =m oblique angles ) ! horizon4 ,,lesson ,,f\r?4 ,,review4 ,read all ! v]tical l9es4 7,,diagram #d47 ,:y is q r a v]tical l9e8 ,:at is a v]tical l9e8 ,read all ! horizontal l9es4 ,:y is #e #f a horizontal l9e8 ,def9e a horizontal l9e4 ,read all ! oblique l9es4 ,:y is s t an oblique l9e4 ,:at is an oblique l9e8 ,,note4--,l9es t po9t 9 ! same direction d n approa* ! same po9t4 ,illu/ration3 ,diagram #d4 ,illu/ration3 ,diagram #e4 ,,lesson ,,fif?4 ,,angles4 ,d ! l9es a b & c d 7,,diagram #e47 po9t 9 ! same direction8 7,see note1 page #ae47 ,!n !y =m an angle ) ea* o!r4 ,:at o!r l9e =ms an angle ) a b8 ,:i* ( ! two l9es c d1 e f1 has ! great] diff]5ce ( direction f ! l9e a b8 ,!n :i* "o =ms ! great] angle ) a b8 ,:at l9e =ms a /ill great] angle ) ! l9e a b8 ,an angle is ! diff]5ce ( direction ( two /raight l9es4 ,if ! l9es a b1 e f1 7 made long]1 w\ld _! direction be *ang$8 ,!n w\ld !re be any great] or less diff]5ce ( direction8 ,!n w\ld ! angles =m$ by !m be any great] or less8 ,!n does ! size ( an angle dep5d ^u ! l5g? ( ! l9es t =m x8 ,if ! l9es a b1 e f1 7 %ort5$1 w\ld ! angle =m$ by !m be any small]8 ,if two l9es =m an angle ) ea* o!r1 & meet1 ! po9t ( meet+ is call$ ! v]tex4 ,:at is ! v]tex ( ! angle =m$ by ! l9es k j1 i j8--i j1 i l8 ,an angle is nam$ by ?ree lett]s1 t :i* d5otes ! v]tex be+ 9 ! middle4 ,?us1 ! angle =m$ by k j1 i j1 is read k j i1 or i j k4 ,read ! f\r angles =m$ by ! l9es m n & o p4 ,! eight =m$ by r s1 t u1 & v w4 ,,lesson ,,six?4 ,,review4 ,read all ! l9es t =m angles ) ! l9e a b4 7,,diagram #f47 ,:i* ( !m =ms ! greate/ angle ) x8 ,illu/ration3 ,diagram #f4 ,:i* ! lea/8 ,( ! two l9es c d1 g h1 :i* =ms ! great] angle ) e f8 ,read all ! angles ^: v]tices are at o on i j4 ,:i* angle is ! great]1 l o m1 or m o j8--i o k1 or i o l8--l o j1 or m o j8 ,read all ! angles =m$ by ! l9es v w & x y4 ,read all ! angles above ! l9e n p4 ,bel[ ! l9e n p4 ,above ! l9e q r4 ,at ! "r ( ! l9e #e u4 ,at ! left4 ,at ! "r ( ! l9e s t4 ,at ! left ( ! l9e s t4 ,:i* angle is ! great]1 n #a #c1 or n #b #d8 ,if ! l9es x y & v w 7 l5g!n$ or produc$1 w\ld ! angles v z x1 y z w be any great]8 ,if !y 7 %ort5$1 w\ld ! angles be any less8 ,:at is an angle8 ,does ! size ( an angle dep5d ^u ! l5g? ( ! l9es :i* =m x8 ,illu/ration3 ,diagram #g4 ,,lesson ,,sev5?4 ,,relations ,,( ,,angles4 ,,adjac5t ,,angles4 ,are ! angles a e c1 c e b 7,,diagram #g471 on ! same side ( any l9e8 ,:at l9e8 ,by :at o!r /raight l9e are !y bo? =m$8 ,!n1 because !y are bo? on ! same side ( ! same /raight l9e a b1 & are bo? =m$ by ! second /raight l9e c d1 !y are call$ 8adjac5t angles40 ,! angles c e b1 b e d are bo? on ! same side ( :at /raight l9e8 ,!y are bo? =m$ by :at second /raight l9e8 ,!n :at k9d ( angles are !y8 ,:y are !y call$ adjac5t angles8 ,read ! adjac5t angles bel[ ! l9e a b4 ,bel[ ! l9e c d4 ,h[ _m pairs ( adjac5t angles c be =m$ by two /raight l9es8 ,read all ! adjac5t angles =m$ by ! l9es l m & n p4 ,,v]tical ,,angles4 ,are ! angles a e c1 b e d =m$ by ! same /raight l9es8 ,are !y adjac5t angles8 ,!y are call$ 8v]tical angles40 ,v]tical angles are angles =m$ by ! same /raight l9es1 b n adjac5t to ea* o!r4 ,read ! o!r pair ( v]tical angles =m$ by ! l9es a b1 c d4 ,read all ! v]tical angles =m$ by ! l9es f g1 i h4 ,by l m1 n p4 ,:y are ! angles l o n1 n o m adjac5t angles8 ,:y are ! angles l o n1 p o m v]tical angles8 ,illu/ration3 ,diagram #h4 ,,lesson ,,eigh?4 ,,review4 ,read ! pairs ( adjac5t angles above ! l9e a b4 7,,diagram #h47 ,:y are !y adjac5t8 ,:at are adjac5t angles8 ,read ! adjac5t angles bel[ ! l9e a b4 ,on ! "r ( ! l9e c d4 ,on ! left4 ,h[ _m pairs ( adjac5t angles are =m$ by ! int]section ( two l9es4 ,read ! pairs ( adjac5t angles =m$ by ! l9es f g & i h4 ,read all ! adjac5t angles =m$ by ! l9es l m1 n p4 ,read all ! pairs ( v]tical angles =m$ by ! l9es a b1 c d4 ,:y are c e b & a e d call$ v]tical angles8 ,:at are v]tical angles8 ,read all ! pairs ( v]tical angles =m$ by ! l9es h i1 f g4 ,h[ _m pairs ( v]tical angles are =m$ by ! int]section ( two l9es8 ,read all ! pairs ( v]tical angles =m$ by ! l9es l m1 n p4 ,,lesson ,,n9?4 ,,k9ds ,,( ,,angles4 ,,"r ,,angles4 ,:at d we call ! angles a o c1 c o b8 7,,diagram #i47 ,are !y equal to ea* o!r8 ,!n !y are call$ "r angles4 ,a "r angle is "o ( two adjac5t angles t are equal to ea* o!r4 ,are ! adjac5t angles c o b1 b o d equal to ea* o!r8 ,!n :at are !y call$8 ,read ! "r angles bel[ ! l9e a b4 ,on ! left ( c d4 ,read ?ree "r angles ^: v]tices are at p4 ,illu/ration3 ,diagram #i4 ,,acute ,,angles4 ,is ! angle m p q great] or less ?an ! "r angle m p r8 ,!n x is call$ an acute angle4 ,an acute angle is "o :i* is less ?an a "r angle4 ,read f\r acute angles ^: v]tices are at p4 ,acute means %>p4 ,:y is r p s an acute angle8 ,:at is an acute angle8 ,,obtuse ,,angles4 ,is ! angle m p s great] or less ?an ! "r angle m p r8 ,!n x is call$ an obtuse angle4 ,an obtuse angle is "o :i* is great] ?an a "r angle4 ,:at o!r obtuse angle has its v]tex at p8 ,obtuse means blunt4 ,read ?ree obtuse angles ^: v]tices are at x4 ,acute & obtuse angles are also call$ oblique angles4 ,,lesson ,,t5?4 ,,review4 ,read all ! "r angles =m$ by ! l9es a b & c d4 7,,diagram #aj47 ,:y are ! adjac5t angles c e b1 b e d1 "r angles8 ,:at is a "r angle8 ,read f\r "r angles ^: v]tices are at n4 ,:i* is ! great]1 ! "r angle p q r1 or ! "r angle t s u8 ,c "o "r angle be great] ?an ano!r8 ,read six acute angles ^: v]tices are at n4 ,:y is m n g an acute angle8 ,:at is an acute angle8 ,:i* is great]1 ! acute angle m n g1 or ! acute angle l n m8 ,may "o acute angle be great] ?an ano!r8 ,:at ?ree acute angles are equal to "o "r angle8 ,illu/ration3 ,diagram #aj4 ,:i* ( ! two acute angles v f w1 y x z is ! great]8 ,read f\r obtuse angles ^: v]tices are at n4 ,:y is f n m an obtuse angle8 ,:at is an obtuse angle8 ,:at does obtuse mean8 ,acute8 ,by :at o!r "n are bo? call$8 ,:i* is great]1 ! l>ge acute angle #a #d #b1 or ! small obtuse angle #a #d #c8 ,h[ mu* great] ?an ! "r angle is ! obtuse angle f n l8 ,h[ mu* less ?an a "r angle is f n i8 ,illu/ration3 ,diagram #aa4 ,,lesson ,,elev5?4 ,,relations ,,( ,,l9es4 ,,p]p5dicul> ,,l9es4 ,:at k9d ( angles d ! l9es a b & c d make ) ea* o!r8 7,,diagram #aa47 ,!n !y are p]p5dicul> to ea* o!r4 ,:at l9e is p]p5dicul> to x y8 ,:y is x p]p5dicul> to x8 ,:at l9e is p]p5dicul> to z #a8 ,:5 is a l9e said to be p]p5dicul> to ano!r8 ,c a l9e /&+ alone be prop]ly call$ a p]p5dicul> l9e8 ,:at two l9es are p]p5dicul> to ! l9es r s8 ,is ! l9e g h p]p5dicul> to ! l9e i j8 ,:y8 ,:at o!r l9e is p]p5dicul> to ! l9e i j8 ,read ?ree l9es t are p]p5dicul> to ! l9e a b4 ,,p>allel ,,l9es4 ,d ! l9es k l1 m n1 diff] 9 direction8 ,!n d !y =m any angle ) ea* o!r8 ,!y are said to be p>allel to ea* o!r4 ,read f\r o!r l9es t are p>allel ) k l4 ,:at l9e is p>allel ) #b #aj8 ,:y8 ,l9es are p>allel ) ea* o!r :5 !y d n diff] 9 direction4 ,,oblique ,,l9es4 ,:at k9d ( angles d ! l9es u t & #h #i =m ) ea* o!r8 ,!n !y are said to be oblique to ea* o!r4 ,l9es are oblique to ea* o!r :5 !y =m oblique angles4 ,see ,note ,c1 ,app5dix4 ,illu/ration3 ,diagram #ab4 ,,lesson ,,twelf?4 ,,review4 ,read five l9es t are p]p5dicul> to ! l9e a b4 7,,diagram #ab47 ,five t are p]p5dicul> to c d4 ,two t are p]p5dicul> to u v1 & meet x4 ,?ree t d n meet x4 ,:y are o p & m n p]p5dicul> to ea* o!r8 ,:5 are l9es said to be p]p5dicul> to ea* o!r8 ,read f\r l9es t are p>allel ) e f4 ,:y are ! l9es e f & g h said to be p>allel to ea* o!r8 ,:5 are l9es said to be p>allel to ea* o!r8 ,read f\r l9es t are p>allel to #e #f4 ,f\r t are p>allel to o p4 ,is any l9e p>allel to u v8 ,c a s+le l9e be prop]ly call$ p]p5dicul>8 ,p>allel8 ,if two l9es are p]p5dicul> to ea* o!r1 :at angle d !y =m8 ,if p>allel1 :at angle8 ,if oblique8 ,illu/ration3 ,diagram #ac4 ,,lesson ,,?irte5?4 ,,relations ,,( ,,angles4 ,,int]ior ,,angles4 ,is ! angle a m n betwe5 ! p>allels1 or outside ( !m8 7,,diagram #ac47 ,x is call$ an int]ior angle4 ,read ?ree o!r int]ior angles betwe5 ! same p>allels4 ,:y is b m n an int]ior angle8 ,an int]ior angle is "o t lies betwe5 p>allel l9es4 ,read ! int]ior angles betwe5 ! p>allel l9es g h & k l4 ,:y is o p l an int]ior angle8 ,:at is an int]ior angle8 ,,ext]ior ,,angles4 ,is ! angle a m e betwe5 ! p>allels1 or outside ( !m8 ,x is call$ an ext]ior angle4 ,read ?ree o!r ext]ior angles =m$ by ! l9es a b1 c d1 & e f4 ,:y is ! angle c n f an ext]ior angle8 ,an ext]ior angle is "o t lies outside ( ! p>allels4 ,,lesson ,,f\rte5?4 ,,review4 ,read all ! int]ior angles =m$ by ! l9es a b1 c d1 & e f4 ,:y is m n d an int]ior angle8 ,:at is an int]ior angle8 ,read all ! ext]ior angles =m$ by ! same l9es4 ,:y is d n f an ext]ior angle8 ,:at is an ext]ior angle8 ,read all int]ior angles =m$ by ! l9es g h1 k l1 & i j4 ,all ! rema9+ int]ior angles 9 ! diagram4 ,all ! ext]ior angles4 ,illu/ration3 ,diagram #ad4 ,,lesson ,,fifte5?4 ,,relations ,,( ,,angles4 ,,opposite ,,angles4 ,are ! angles e m b1 b m n1 on ! same side ( ! int]sect+ l9e e f8 ,are !y adjac5t8 ,are e m b1 m n d1 on ! same side ( ! int]sect+ l9e e f8 ,are !y adjac5t8 ,!n !y are call$ opposite angles4 ,opposite angles lie on ! same side ( ! int]sect+ l9e1 b are n adjac5t4 ,are ! angles e m b1 f n d1 on ! same side ( ! int]sect+ l9e8 ,are !y adjac5t8 ,!n are !y opposite8 ,are !y int]ior or ext]ior angles8 ,!n !y are 8opposite ext]ior angles40 ,:y are !y ext]ior8 ,:y are !y opposite8 ,are ! angles b m n1 m n d1 opposite angles8 ,are !y int]ior or ext]ior angles8 ,!n !y are 8opposite int]ior angles40 ,:y are !y opposite8 ,:y int]ior8 ,read ! opposite ext]ior angles on ! left ( ! l9e e f4 ,read ! opposite int]ior angles on ! same side4 ,are ! opposite angles e m a1 m n c1 bo? ext]ior or int]ior8 ,!n !y are opposite ext]ior & int]ior angles4 ,read two pairs ( opposite ext]ior & int]ior angles on ! "r ( e f4 ,on ! left4 ,,alt]nate ,,angles4 ,d ! angles b m n1 m n c1 lie on ! same side ( ! int]sect+ l9e e f8 ,are !y adjac5t to ea* o!r8 ,are !y v]tical angles8 ,!n !y are alt]nate angles4 ,alt]nate angles lie on diff]5t sides ( ! int]sect+ l9e1 & are nei!r adjac5t nor v]tical4 ,are ! alt]nate angles b m n1 m n c1 ext]ior or int]ior8 ,!n !y are call$ 8int]ior alt]nate angles40 ,read ano!r pair ( int]ior alt]nate angles betwe5 a b & c d4 ,are ! angles e m b1 c n f1 alt]nate angles8 ,:y8 ,are !y ext]ior or int]ior8 ,!n :at may !y be call$8 ,read ano!r pair ( ext]ior alt]nate angles4 ,:y are e m a1 d n f1 alt]nate angles8 ,:y ext]ior alt]nate8 ,,lesson ,,sixte5?4 ,,review4 ,read ! ext]ior opposite angles on ! "r ( ! l9e e f4 7,,diagram #ad47 ,on ! left4 ,on ! "r ( r s4 ,on ! left4 ,:y are e m a1 c n f1 ext]ior angles8 ,:y are !y opposite angles8 ,:at are opposite angles8 ,read ! int]ior opposite angles on ! "r ( ! int]sect+ l9e e f4 ,on ! left ( x4 ,on ! "r ( r s4 ,on ! left4 ,read ! int]ior alt]nate angles =m$ by ! l9es a b1 c d1 & e f4 ,:i* pair are acute angles8 ,:i* pair are obtuse angles8 ,:y are b m n1 m n c1 int]ior angles8 ,:y alt]nate8 ,:at are alt]nate angles8 ,read ! ext]ior alt]nate angles ( ! same l9es4 ,read ! acute int]ior alt]nate angles ( ! p>allels t u1 v w4 ,! obtuse4 ,! acute ext]ior alt]nate angles4 ,obtuse4 ,read ! pair ( opposite ext]ior angles on ! "r ( ! l9e e f4 ,on ! left4 ,on ! "r ( r s4 ,on ! left4 ,illu/ration3 ,diagram #ae4 ,,lesson ,,sev5te5?4 ,,review4 ,read ?irte5 or m angles ^: v]tices are at c4 7,,diagram #ae47 ,read f\r obtuse angles4 ,read two "r angles4 ,:at ?ree acute angles equal "o "r angle8 ,:i* is great]1 ! "r angle #d1 or ! "r angle #e8 ,! obtuse angle #f1 or ! acute angle #g8 ,read twelve pairs ( adjac5t angles =m$ by ! l9es w x1 c4 ,read six pairs ( v]tical angles =m$ by ! same l9es4 ,read all ! int]ior angles =m$ by ! l9es i j1 k l1 & m n4 ,read all ! ext]ior angles =m$ by ! same l9es4 ,two pairs ( opposite ext]ior angles4 ,two pairs ( opposite int]ior angles4 ,f\r pairs ( opposite ext]ior & int]ior angles4 ,two pairs ( alt]nate int]ior angles4 ,two pairs ( alt]nate ext]ior angles4 ,:y are ! angles i o m1 m o j1 call$ adjac5t8 ,:at are adjac5t angles8 ,:at k9d ( an angle is i o m8 ,:y8 ,:at is an acute angle8 ,:at k9d ( an angle is m o j8 ,:y8 ,:at is an obtuse angle8 ,:y are a c f1 f c b1 "r angles8 ,:at is a "r angle8 ,:y are m o i1 j o p1 v]tical angles8 ,:at are v]tical angles8 ,:y is m o i an ext]ior angle8 ,:at is an ext]ior angle8 ,:y is j o p an int]ior angle8 ,:at is an int]ior angle8 ,:y are m o i1 o p k1 opposite angles8 ,:at are opposite angles8 ,:y are j o p1 o p k1 alt]nate angles8 ,:at are alt]nate angles8 ,,lesson ,,eighte5?4 ,,problems4 ,draw an obtuse angle :i* %all be only a little l>g] ?an a "r angle4 ,draw "o :i* %all be mu* great] ?an a "r angle4 ,draw an acute angle :i* %all be only a little less ?an a "r angle4 ,draw "o :i* %all be mu* less ?an a "r angle4 ,draw an obtuse angle ) l9es ab\t "o in* long4 ,draw an acute angle ) sides ?ree in*es long4 ,:i* is great]1 ! obtuse angle1 or ! acute angle8 ,draw a "r angle ) l9es an in* long4 ,draw "o ) l9es five in*es long4 ,:i* is ! great]1 fir/ or ! second8 ,illu/ration3 ,diagram #af4 ,,lesson ,,n9ete5?4 ,,polygons4 ,"n any ?+ besides y\r desk t has a flat surface4 ,a flat surface is call$ a plane4 ,h[ _m sides has ! plane ,fig4 ,a8 7,,diagram #af47 ,x is call$ a triangle4 8,tri0 means 8?ree40 ,:at o!r triangles d y see4 ,triangles are sometimes call$ trigons4 ,a triangle is a plane figure hav+ ?ree sides4 ,h[ _m sides has ! plane figure m>k$ ,b8 ,h[ _m angles8 ,x is call$ a quadrangle1 or quadrilat]al4 8,quad0 d5otes 8f\r40 ,:at o!r quadrangles d y see8 ,:y is ,fig4 ,b a quadrangle8 ,a quadrangle is a plane figure hav+ f\r sides4 ,h[ _m sides has ! ,fig4 ,c8 ,x is call$ a p5tagon4 ,:at o!r p5tagon d y see8 ,:y is ,fig4 ,c a p5tagon8 ,a p5tagon is a plane figure hav+ five sides4 ,9 l mann]1-- ,a hexagon is a plane figure hav+ six sides4 ,a heptagon is a plane figure hav+ sev5 sides4 ,an octagon has eight sides4 ,a nonagon has n9e sides4 ,a decagon has t5 sides4 ,all ^! figures are call$ polygons4 8,poly0 means 8_m40 ,:at d y call a polygon ( ?ree sides8 ,( f\r sides8 ,( six sides8 c4 ,if ! l5g? ( ea* side ( triangle ,a is "o in*1 h[ long are ! ?ree sides toge!r8 ,! sum ( ! sides ( a polygon is its p]imet]4 ,:i* ( ! triangles has unequal sides8 ,:i* has equal sides8 ,! latt] is call$ a regul> polygon4 ,:i* p5tagon has "o side long] ?an any "o ( its o!r sides8 ,:i* has its sides all equal to ea* o!r8 ,are its angles also equal8 ,x is !re=e a regul> polygon1 or regul> p5tagon4 ,"n a hexagon t is n regul>4 ,"n a regul> hexagon4 ,a regul> octagon4 ,a regul> heptagon4 ,a polygon is a plane figure b\nd$ by /raight l9es4 ,,lesson ,,tw5tie?4 ,,review4 ,"n all ! triangles4 7,,diagram #af47 ,:y is ,fig4 ,a a triangle8 ,:at is a triangle8 ,:at o!r "n is sometimes giv5 to triangles8 ,"n all ! quadrilat]als4 ,:y is ,fig4 ,b a quadrilat]al8 ,:at is a quadrilat]al1 or quadrangle8 ,"n all ! p5tagons1 hexagons1 heptagons1 octagons1 & nonagons4 ,:y is ,c a p5tagon8 ,:at is a p5tagon8 ,a hexagon8 ,a heptagon8 c4 ,h[ _m polygons 9 ! diagram8 ,:at is a polygon8 ,if ea* side ( ,fig4 ,b is "o in*1 h[ _m in*es are !re 9 its p]imet]8 ,:5 is a polygon regul>8 ,"n all ! regul> polygons 9 diagram #af4 ,"n all ! irregul> polygons4 ,illu/ration3 ,diagram #ag4 ,,lesson ,,tw5ty-,,fir/4 ,,triangles4 ,,acute-,,angl$ ,,triangles4 ,9 ! triangle #a1 :at k9d ( an angle is b a c8 a c b8 c b a8 7,,diagram #ag47 ,!n x is call$ an acute-angl$ triangle4 ,an acute-angl$ triangle is "o ^: angles are all acute4 ,read ?ree o!r acute-angl$ triangles4 ,,obtuse-,,angl$ ,,triangles4 ,9 ! triangle #d1 :at k9d ( an angle is l k m8 ,!n x is call$ an obtuse-angl$ triangle4 ,an obtuse-angl$ triangle is "o t has "o obtuse angle4 ,"n two o!rs4 ,,"r-,,angl$ ,,triangles4 ,9 ! triangle #c1 :at k9d ( an angle is g i j8 ,!n x is call$ a "r-angl$ triangle4 ,a "r-angl$ triangle is "o t has "o "r angle4 ,"n ?ree o!r "r-angl$ triangles4 ,^u :i* side does ! triangle #c seem to /&8 ,!n i j is call$ ! base ( ! triangle4 ,:at lett] m>ks ! v]tex ( ! angle opposite ! base8 ,!n ! po9t g is said to be ! v]tex ( ! triangle4 ,if1 9 ! triangle #g1 we consid] t v ! base1 :at po9t is ! v]tex8 ,if v be consid]$ ! v]tex1 :i* side w be ! base8 ,9 ! triangle #c1 :at side is opposite ! "r angle8 ,!n g j is call$ ! hypo!nuse ( ! triangle4 ,! hypo!nuse ( a triangle is ! side opposite ! "r angle4 ,read ! hypo!nuse ( ea* ( ! triangles #e1 #f1 & #aa4 ,ei!r side ab\t ! "r angle may be consid]$ ! base4 ,!n ! o!r side w be ! p]p5dicul>4 ,9 ! triangle #c1 if i j is ! base1 :i* side is ! p]p5dicul>8 ,if g i be consid]$ ! base1 :i* side is ! p]p5dicul>8 ,9 triangle #e1 if n o is ! base1 :i* side is ! p]p5dicul>8 ,,lesson ,,tw5ty-,,second4 ,,review4 ,"n f\r acute-angl$ triangles4 7,,diagram #ag47 ,:y is ! triangle #h acute-angl$8 ,:at is an acute-angl$ triangle8 ,"n ?ree obtuse-angl$ triangles4 ,:y is ! triangle #i an obtuse-angl$ triangle8 ,:at is an obtuse-angl$ triangle8 ,"n f\r "r-angl$ triangles4 ,:y is ! triangle #f a "r-angl$ triangle8 ,:at is a "r-angl$ triangle8 ,9 ! triangle #f1 :i* side is ! hypo!nuse8 ,:y8 ,:at is ! hypo!nuse8 ,:at two sides ( ! triangle #f may be reg>d$ z ! base8 ,if q r be consid]$ ! base1 :at d y call ! side q s8 ,read ! hypo!nuse ( ea* ( ! triangles #c1 #e1 #f1 & #aa4 ,illu/ration3 ,diagram #ah4 ,,lesson ,,tw5ty-,,?ird4 ,,triangles4 7,cont9u$47 ,,isosceles ,,triangles4 ,( ! triangle #a1 :i* two sides are equal to ea* o!r8 ,!n x is call$ an isosceles triangle4 ,an isosceles triangle is "o t has two equal sides4 ,"n eight isosceles triangles4 ,:y is ! triangle #b an isosceles triangle8 ,:at k9d ( a triangle is x on acc\nt ( its angles8 ,!n x is an acute-angl$ isosceles triangle4 ,"n f\r acute-angl$ isosceles triangles4 ,:at k9d ( a triangle is ,fig4 #d on acc\nt ( ! angle k j l8 ,:at k9d on acc\nt ( its equal sides8 ,!n x is call$ an obtuse-angl$ isosceles triangle4 ,"n "o o!r obtuse-angl$ isosceles triangle4 ,:at k9d ( a triangle is ,fig4 #f on acc\nt ( ! angle q p r8 ,:at k9d on acc\nt ( its equal sides8 ,!n x is call$ a "r-angl$ isosceles triangle4 ,"n "o o!r "r-angl$ isosceles triangle4 ,:y is ,fig4 #ab a "r-angl$ triangle8 ,:y isosceles8 ,,equilat]al ,,triangles4 ,:i* ( ! isosceles triangles has all its ?ree sides equal to ea* o!r8 ,x is call$ an equilat]al triangle4 8,equi0 means 8equal40 8,latus0 means a 8side40 ,an equilat]al triangle is "o t has its ?ree sides equal to ea* o!r4 ,:at k9d ( a triangle is ,fig4 #g on acc\nt ( its ?ree equal sides8 ,:at k9d on acc\nt ( its two equal sides s t1 s u1 or t s1 t u1 or u s1 u t8 ,!n mu/ n e equilat]al triangle be also isosceles8 ,:at k9d ( a triangle is ,fig4 #b on acc\nt ( its equal sides d e1 d f8 ,if ! side e f is long] ?an ei!r ( ! o!r two sides1 is x an equilat]al triangle8 ,!n is e isosceles triangle also equilat]al8 ,"n ano!r isosceles triangle t is n equilat]al4 ,"n "o t is equilat]al4 ,9 any equilat]al triangle ! ?ree angles are equal to ea* o!r4 ,on acc\nt ( its equal angles1 x is also call$ an equiangul> triangle4 ,:at is ,fig4 #h call$ on acc\nt ( its ?ree equal sides8 ,on acc\nt ( its ?ree equal angles8 ,e equilat]al triangle is also equiangul>4 ,e equiangul> triangle is also equilat]al4 ,"n a triangle t has no two sides equal to ea* o!r4 ,x is call$ a scal5e triangle4 ,:at k9d ( a triangle is ,fig4 #e on acc\nt ( its "r angle8 ,:at k9d on acc\nt ( its ?ree unequal sides8 ,!n x is a "r-angl$ scal5e triangle4 ,:at "n c y give ,fig4 #aa on acc\nt ( ! angle g e f8 ,on acc\nt ( its ?ree unequal sides8 ,!n :at may x be call$8 ,illu/ration3 ,diagram #ai4 ,,lesson ,,tw5ty-,,f\r?4 ,,review4 ,"n eight isosceles triangles4 7,,diagram #ai47 ,:y is ,fig4 #b an isosceles triangle8 ,:at is an isosceles triangle8 ,"n two "r-angl$ isosceles triangles4 ,"n five acute-angl$ isosceles triangles4 ,"n "o obtuse-angl$ isosceles triangle4 ,"n two isosceles triangles t are also equilat]al4 ,are all isosceles triangles equilat]al8 ,"n six isosceles triangles t are n equilat]al4 ,:at does 8equi0 mean8 8,latus08 ,:at are equilat]al triangles call$ on acc\nt ( _! equal angles8 ,are all equilat]al triangles equiangul>8 ,are all equiangul> triangles equilat]al8 ,:at are equilat]al triangles8 ,"n f\r scal5e triangles4 ,"n two "r-angl$ scal5e triangles4 ,:y is ,fig4 #c a "r-angl$ triangle8 ,:y scal5e8 ,:at is a scal5e triangle8 ,"n "o obtuse-angl$ scal5e triangle4 ,"n "o acute-angl$ scal5e triangle4 ,,problems4 ,f ! same po9t draw two /raight l9es ( any l5g?1 mak+ an acute angle ) ea* o!r4 ,make !m equal to ea* o!r by measur+4 ,jo9 _! ends4 ,:at k9d ( a triangle is x on acc\nt ( its angles8 ,on acc\nt ( its two equal sides8 ,write its two names inside ( x4 ,draw an isosceles triangle ^: equal sides %all ea* be less ?an ! ?ird side4 ,write its two names )9 x4 ,draw an oblique /raight l9e twice z long z any %ort measure or unit4 ,at "o end draw a /raight l9e p]p5dicul> to x1 & ?ree times z long z ! same measure4 ,connect ! ends ( ! two l9es by a /raight l9e4 ,:at k9d ( an angle is t opposite ! la/ l9e drawn8 ,are any two ( its sides equal8 ,write its two names "u x4 ,draw a horizontal /raight l9e ( any l5g?4 ,at "o end draw a v]tical l9e ( equal l5g?4 ,complete ! triangle1 & write two names inside4 ,draw a "r-angl$ triangle ^: base is ( any l5g?1 & its p]p5dicul> twice z long4 ,draw a "r-angl$ triangle ^: base is ?ree times z long z any %ort measure1 & its p]p5dicul> five times z long z ! same measure or unit4 ,illu/ration3 ,diagram #bj4 ,,quadrilat]als4 ,h[ _m sides has ! figure a b d c8 ,:at is x call$ on acc\nt ( ! numb] ( its sides8 ,"n ?ree o!r quadrilat]als ^: v]tices are m>k$4 ,"n sev5 by numb]s4 ,quadrilat]als are sometimes nam$ by means ( two opposite v]tices4 ,! quadrilat]al a b d c1 or c d b a1 may be read a d1 or b c1 or c b1 or d a4 ,"n ! quadrilat]al1 g h f e1 f\r ways4 ,h[ _m angles has ea* figure8 ,on acc\nt ( ! numb] ( _! angles !y are call$ quadrangles4 ,has ! quadrilat]al a d any two sides p>allel to ea* o!r8 ,!n x is call$ a trapezium4 ,a trapezium is a quadrilat]al t has no two sides p>allel4 ,"n two o!r trapeziums4 ,:y is ,fig4 #g a trapezium8 ,has ! quadrilat]al e h any two sides p>allel8 ,:i* two8 ,are ! o!r two sides p>allel8 ,x is call$ a 8trapezoid40 8,oid0 means l4 ,:at does 8trapezoid0 mean8 ,a trapezoid is a quadrilat]al t has only "o pair ( sides p>allel4 ,"n ano!r trapezoid4 ,:y is ,fig4 #f a trapezoid8 ,h[ _m pairs ( p>allel sides has ! quadrilat]al i l8 ,"n ! horizontal p>allels4 ,"n ! oblique p>allels4 ,x is call$ a 8p>allelogram40 ,a p>allelogram is a quadrilat]al ^: opposite sides are p>allel4 ,"n five o!r p>allelograms4 ,:y is ,fig4 #d a p>allelogram8 ,:y is n ,fig4 #f a p>allelogram8 ,:y is n e h a p>allelogram8 ,:at two names may y give to ,fig4 #e8 ,:y is x a quadrilat]al8 ,:y a trapezium8 ,:at two names may we give to ,fig4 #f8 ,:y is x a quadrilat]al8 ,:y a trapezoid8 ,:at two names may we give to ,fig4 #c8 ,:y is x a p>allelogram8 ,:y a quadrilat]al8 ,,lesson ,,tw5ty-,,fif?4 ,,review4 ,h[ _m quadrilat]als 9 ! diagram4 7,,diagram #bj47 ,:y is ,fig4 a d a quadrilat]al8 ,:at is a quadrilat]al8 ,on acc\nt ( ! numb] ( its angles1 :at may x be call$8 ,"n all ! quadrilat]als4 ,"n ?ree trapeziums4 ,:y is ,fig4 #e a trapezium8 ,:at is a trapezium8 ,"n two trapezoids4 ,:y is ,fig4 #f a trapezoid8 ,"n its p>allel sides4 ,:at is a trapezoid8 ,"n six p>allelograms4 ,:y is ,fig4 #d a p>allelogram8 ,"n its two pairs ( p>allel sides4 ,:at is a p>allelogram8 ,:at two names c y give to ,fig4 #d8 ,:y ! fir/8 ,:y ! second8 ,:at two names may be giv5 to ,fig4 #g8 ,:y ! fir/8 ,:y ! second8 ,:at two to ,fig4 #f8 ,:y ! fir/8 ,:y ! second8 ,illu/ration3 ,diagram #ba4 ,,lesson ,,tw5ty-,,six?4 ,,k9ds ,,( ,,p>allelograms4 ,,rhomboids4 ,h[ _m quadrilat]als 9 ! diagram8 7,,diagram #ba47 ,h[ _m p>allelograms8 ,has ! p>allelogram a d any "r angle8 ,x is call$ a 8rhomboid40 ,a rhomboid is a p>allelogram :i* has no "r angle4 ,"n five o!r rhomboids4 ,:at ?ree names may be giv5 to ,fig4 #b8 ,:y is x a quadrilat]al8 ,:y a p>allelogram8 ,:y a rhomboid8 ,,rhombs4 ,are ! f\r sides ( ! rhomboid a d equal to ea* o!r8 ,are ! f\r sides ( ! rhomboid e h equal to ea* o!r8 ,if a triangle has its ?ree sides equal to ea* o!r1 :at d y call x8 ,!n :5 a rhomboid has its sides equal to ea* o!r1 :at may x be call$8 ,an equilat]al rhomboid is call$ a rhombus4 ,a rhombus is an equilat]al rhomboid4 ,see ,note ,d1 ,app5dix4 ,"n two o!r rhombuses1 or rhombs4 ,:at f\r names c y give to ,fig4 e h8 ,:y a quadrilat]al8 ,:y a p>allelogram8 ,:y a rhomboid8 ,:y a rhombus8 ,,rectangles4 ,has ! p>allelogram i l any "r angles8 ,h[ _m8 ,x is call$ a 8rectangle40 ,a rectangle is a "r-angl$ p>allelogram4 ,"n f\r o!r rectangles4 ,:at ?ree names may be giv5 to ,fig4 i l8 ,:y a quadrilat]al8 ,:y a p>allelogram8 ,:y a rectangle8 ,,squ>es4 ,has ! rectangle i l its f\r sides equal8 ,has ! rectangle m p its f\r sides equal8 ,x is call$ a 8squ>e40 ,a squ>e is an equilat]al rectangle4 ,"n ano!r 8squ>e40 ,:at f\r names may be giv5 to ,fig4 m p8 ,:y a quadrilat]al8 ,:y a p>allelogram8 ,:y a rectangle8 ,:y a squ>e8 ,,lesson ,,tw5ty-,,sev5?4 ,,review4 ,"n six rhomboids4 7,,diagram #ba47 ,:at ?ree names may be giv5 to ,fig4 #c8 ,:y a quadrilat]al8 ,:y a p>allelogram8 ,:y a rhomboid8 ,:at is a quadrilat]al8 ,p>allelogram8 ,rhomboid8 ,"n ?ree rhombs4 ,:at f\r names may y give ,fig4 #e8 ,:y a quadrilat]al8 ,:y a p>allelogram8 ,:y a rhomboid8 ,:y a rhomb8 ,:at is a rhomboid8 ,a rhomb8 ,"n five rectangles4 ,:at ?ree names may be giv5 to ,fig4 #a8 ,:y a quadrilat]al8 ,:y a p>allelogram8 ,:y a rectangle8 ,:at is a rectangle8 ,"n two squ>es8 ,by :at f\r names may ,fig4 #g be call$8 ,:y by ! fir/8 ,by ! second8 ,by ! ?ird8 ,by ! f\r?8 ,:at is a squ>e8 ,:at is a rectangle8 ,:at is a p>allelogram8 ,:at is a quadrilat]al8 ,illu/ration3 ,diagram #bb4 ,,lesson ,,tw5ty-,,eigh?4 ,,comp>ison ,,& ,,contra/4 ,,trapezium ,,& ,,trapezoid4 ,9 :at respect are ,figs4 ,a & ,b alike8 ,on ?is acc\nt1 :at "n may be giv5 to ea*8 ,h[ does ,fig4 ,b diff] f ,fig4 ,a8 ,:at p>ticul> "n may y give to ,fig4 ,b8 ,:at "o to ,fig4 ,a8 ,,rhomboid ,,& ,,rectangle4 ,9 :at two respects are ,figs4 ,c & ,d alike8 ,on acc\nt ( ! numb] ( _! sides1 :at may ea* be call$8 ,because _! opposite sides are p>allel1 :at may ea* be call$8 ,9 :at respect d !y diff]8 ,:at p>ticul> "n may be giv5 to ,fig4 ,c8 ,:at "o to ,fig4 ,d8 ,:at ?ree names may y give to ! figure ) "r angles8 ,:at ?ree to ! "o )\t "r angles8 ,,rhomboid ,,& ,,rhombus4 ,9 :at ?ree ?+s are ,figs4 ,e & ,f alike8 ,:at ?ree names may be giv5 to ea*8 ,h[ d !y diff] f ea* o!r8 ,:at p>ticul> "n may y give to ,fig4 ,f8 ,:at f\r names has ,fig4 ,f8 ,,rectangle ,,& ,,squ>e4 ,9 :at ?ree ?+s are ,figs4 ,g & ,h alike8 ,on acc\nt ( ! numb] ( _! sides1 :at may ea* be call$8 ,because _! opposite sides are p>allel1 :at may ea* be call$8 ,because !y h "r angles1 :at may !y be call$8 ,9 :at respect is ,fig4 ,h diff]5t f ,fig4 ,g8 ,on ?is acc\nt1 :at p>ticul> "n may be appli$ to ,fig4 ,h8 ,:at ?ree names may be appli$ to ,fig4 ,g8 ,:at f\r to ,fig4 ,h8 ,,rhombus ,,& ,,squ>e4 ,9 :at ?ree ?+s are ,figs4 ,f & ,h alike8 ,on acc\nt ( ! numb] ( _! sides1 :at "n may be giv5 to ea*8 ,because _! opposite sides are p>allel1 :at "n may be giv5 to ea*8 ,because bo? are p>allelograms1 & bo? h _! sides equal1 :at "n may be giv5 to ea*8 ,:at p>ticul> "n has ,fig4 ,f8 ,:at p>ticul> "n has ,fig4 ,h8 ,:at f\r names may be giv5 to ,fig4 ,f8 ,:at f\r to ,fig4 ,h8 ,,lesson ,,tw5ty-,,n9?4 ,,review4 ,:at two names may be giv5 to ,fig4 ,a4 7,,diagram #bb47 ,to ,fig4 ,b8 ,9 :at are !y alike8 ,9 :at d !y diff]8 ,by :at ?ree names may ,fig4 ,c be call$8 ,by :at ?ree names may ,fig4 ,d be call$8 ,9 :at two ?+s are !y alike8 ,9 :at "o ?+ d !y diff]8 ,:at p>ticul> "n has ,c8 ,:at "o has ,d8 ,:at ?ree names may be appli$ to ,fig4 ,e8 ,:at f\r to ,fig4 ,f8 ,:at prop]ty has ,f t ,e has n8 ,:at p>ticul> "n has x on t acc\nt8 ,:at ?ree names has ,fig4 ,g8 ,:at f\r has ,fig4 ,h8 ,:at prop]ty has ,fig4 ,h t ,g has n8 ,:at p>ticul> "n has x 9 consequ5ce8 ,:at f\r names may y give to ,fig4 ,f8 ,:at f\r to ,fig4 ,h8 ,:at ?ree names may be appli$ to ei!r8 ,9 :at ?ree ?+s are !y alike8 ,9 :at respect d !y diff]8 ,:at p>ticul> "n has ,fig4 ,f8 ,:at p>ticul> "n has ,fig4 ,h8 ,illu/ration3 ,diagram #bc4 ,,lesson ,,?irtie?4 ,,measurem5t ,,( ,,surfaces4 ,9 ,fig4 #a 7,,diagram #bc47 call ! l9e a b a unit4 ,rectangle #a is h[ _m units long8 ,h[ _m high8 ,because its sides are equal1 :at is x call$8 ,rectangle #b is h[ _m units long8 ,h[ _m high or wide8 ,h[ _m squ>es does x conta98 ,rectangle #c is h[ _m units long8 ,h[ _m wide8 ,h[ _m squ>es does x conta98 ,if x 7 f\r units long & "o wide1 h[ _m squ>es w\ld x conta98 ,if x 7 five long & "o wide8 ,six long8 c4 ,rectangle #d is h[ _m units long8 ,h[ _m wide8 ,h[ _m squ>es does x conta98 ,h[ _m squ>es 9 t "p :i* is two units long1 m n1 & "o unit wide1 m l8 ,on acc\nt ( ! second unit 9 wid?1 l k1 h[ _m times two squ>es are !re8 ,if ! wid? 7 "o unit m1 h[ _m times two squ>es w\ld !re be8 ,rectangle #e is h[ _m units long8 ,h[ _m units wide or high8 ,h[ _m squ>es does x conta98 ,h[ _m squ>es 9 t "p :i* is ?ree units long1 o p1 & "o unit wide1 o t8 ,! second unit 9 wid?1 t q1 gives h[ _m m squ>es8 ,h[ _m times ?ree squ>es8 ,if ano!r unit 7 add$ to ! wid?1 h[ _m m squ>es w\ld be made8 ,h[ _m times ?ree squ>es8 ,if x 7 f\r units wide1 h[ _m times ?ree squ>es w\ld !re be8 ,rectangle #f is h[ _m units long8 ,h[ _m units high or wide8 ,h[ _m squ>es 9 t "p :i* is f\r units long & "o high8 ,h[ _m times f\r squ>es 9 t "p :i* is f\r long & two high8 ,h[ _m times f\r squ>es :5 x is f\r long & ?ree high8 ,if ano!r unit 7 add$ to ! height1 h[ _m m squ>es w\ld be add$8 ,h[ _m times f\r squ>es w\ld !re be8 ,if a rectangle 7 five units long & "o unit wide1 h[ _m squ>e units w\ld x conta98 ,if x 7 two units wide1 h[ _m times five squ>e units w\ld x conta98 ,if x 7 ?ree units wide8 ,f\r8 c4 ,if y\r rul] is t5 in*es long & only "o in* wide1 h[ _m squ>e in*es are !re 9 x8 ,if x 7 two in*es wide1 h[ _m times t5 squ>e in*es w\ld x conta98 ,if y\r ari?metic-cov] is sev5 in*es long & five in*es wide1 h[ _m squ>e in*es are !re 9 x8 ,if a wall ( ?is room is tw5ty feet long1 h[ _m squ>e feet are !re 9 t "p :i* is "o foot high8 ,two high8 ,?ree high8 ,f\r high8 ,if ! same wall is sixte5 feet high1 h[ _m squ>e feet 9 x8 ,fig4 #e has h[ _m times ?ree squ>es8 ,fig4 #g has h[ _m times two squ>es8 ,:i* has ! great] numb] ( squ>es8 ,:at diff]5ce is !re betwe5 two times ?ree squ>es & ?ree times two squ>es8 ,,lesson ,,?irty-,,fir/4 ,,review4 ,draw a rectangle ( any wid? ^: l5g? is ?ree times ! wid?4 ,h[ _m squ>es has x if ! wid? be tak5 z ! unit8 ,make x twice z wide z be=e4 ,h[ _m squ>es has x n[8 ,:at two numb]s multipli$ toge!r w give ! numb] ( squ>es8 ,make x ?ree times z wide4 ,h[ _m squ>es has x n[8 ,:at two numb]s multipli$ toge!r w give ! numb] ( squ>es8 ,! cov] ( a geography is "o foot long & "o foot wide1 h[ _m squ>e feet 9 x8 ,h[ _m in*es long is ! same cov]8 ,h[ _m wide8 ,h[ _m squ>e in*es does x conta98 ,h[ _m squ>e in*es are equal to "o squ>e foot8 ,a table is "o y>d long & "o y>d wide1 h[ _m squ>e y>ds 9 x8 ,h[ _m feet long is ! same table8 ,h[ _m feet wide8 ,h[ _m squ>e feet does x conta98 ,"o squ>e y>d equals h[ _m squ>e feet8 ,draw a squ>e ^: side is a unit ( any l5g?4 ,draw ano!r ^: side is two units ( ! same l5g?4 ,! second squ>e is h[ _m times z l>ge z ! fir/ "o8 ,h[ _m squ>es 9 half ! second squ>e8 ,:i* is great]1 two squ>e in*es1 or two in*es squ>e8 ,two in*es squ>e is h[ _m times two squ>e in*es8 ,draw a squ>e ^: side is ?ree in*es4 ,h[ _m squ>e in*es does x conta98 ,h[ _m times z _m squ>es z ! squ>e ( "o in*8 ,h[ _m squ>e in*es 9 ! bottom r[8 ,h[ _m 9 all8 ,:i* is great]1 ?ree in*es squ>e1 or ?ree squ>e in*es8 ,?ree in*es squ>e is h[ _m times ?ree squ>e in*es8 ,,problems4 ,an equilat]al triangle has ea* ( its sides "o in* long1 :at is its p]imet]8 ,if ea* side 7 two in*es long1 :at w\ld be its p]imet]8 ,an isosceles triangle has its two equal sides ea* ?ree in*es long1 & its ?ird side five in*es long1 :at is its p]imet]8 ,a "r-angl$ isosceles triangle has its base five in*es1 & its hypo!nuse sev5 in*es long1 :at is its p]imet]8 ,a squ>e geography-cov] is n9e in*es long on "o side1 h[ long all r\nd8 ,h[ _m squ>e in*es 9 x8 ,a slate is sixte5 in*es long & twelve wide1 h[ _m in*es all r\nd x8 ,a rectangle is five in*es long & ?ree wide1 h[ long all r\nd8 ,h[ _m squ>e in*es 9 x8 ,a slate is "o foot long & eight in*es wide1 :at is its p]imet]8 ,a room is tw5ty-f\r feet long & tw5ty-"o feet wide1 h[ _m feet all r\nd x8 ,h[ _m squ>e feet 9 ! floor8 ,h[ _m pieces ( pap] ea* a foot squ>e w\ld exactly cov] x8 ,a y>d ( c>pet is two feet wide1 h[ _m squ>e feet 9 x8 ,*>les & ,h5ry />t f ! same place1 & walk 9 opposite directions2 ,*>les goes tw5ty y>ds1 & ,h5ry fifte51 h[ _m y>ds ap>t are !y8 ,if !y />t f opposite ends ( a /raight walk tw5ty-five feet long1 & walk t[>ds ea* o!r1 h[ _m feet w ,*>les h to walk to meet ,h5ry :o has walk$ fifte5 feet8 ,a lot is =ty rods long & ?irty wide1 h[ long mu/ ! f5ce be8 ,:at l5g? ( f5ce w divide x into f\r equal p>ts8 ,illu/ration3 ,diagram #bd4 ,,lesson ,,?irty-,,second4 ,,! ,,circle ,,& ,,its ,,l9es4 ,if ! /raight l9e c a 7 a /r+ made fa/ at c1 ) a %>p p5cil-po9t at ! o!r end a1 & ! p5cil-po9t 7 mov$ t[>ds d1 :at l9e w\ld be drawn8 ,:at k9d ( a l9e w\ld x be8 ,if ! p5cil-po9t cont9u$ to move 9 ! same direction until x return$ to ! />t+-po9t a1 :at curv$ l9e w\ld be drawn1 nam+ x by all ! po9ts 9 x :i* are m>k$8 ,! plane figure b\nd$ by ?is curve is call$ a 8circle40 ,:at po9t is at ! c5tre ( ?is figure8 ,a circle is a plane figure b\nd$ by a curv$ l9e1 all po9ts ( :i* are equally di/ant f ! c5tre4 ,! curv$ l9e is call$ a 8circumf]5ce40 ,! circumf]5ce ( a circle is ! curve :i* b\nds x4 ,"n a /raight l9e t jo9s two po9ts 9 ! circumf]5ce4 ,x is call$ a 8*ord40 ,a *ord is a /raight l9e t jo9s two po9ts ( a circumf]5ce4 ,read six *ords 9 ! diagram4 ,:i* two ( ^! *ords pass ?r\gh ! c5tre8 ,!y are call$ 8diamet]s40 ,a diamet] is a *ord t passes ?r\gh ! c5tre4 ,"n a l9e t jo9s ! c5tre ) a po9t ( ! circumf]5ce4 ,x is call$ a 8radius40--7,plural1 radii47 ,a radius is a /raight l9e t jo9s ! c5tre to a po9t ( ! circumf]5ce4 ,read five radii4 ,:i* is f>!r f ! c5tre1 ! po9t a or ! po9t d8 ,c ! radius c d be great] ?an ! radius c a8 ,or great] ?an c v1 or c o8 ,!n all radii ( ! same circle are equal to ea* o!r4 ,:at d we call ! l9es o d1 c d1 c o8 ,:at "p ( ! diamet] o d is ! radius o c8 ,"n a *ord t is produc$ )\t ! circle4 ,x is call$ a 8secant40 ,a secant is a *ord produc$4 ,"n two secants4 ,if ! *ord d i 7 made a secant1 w\ld x become long] or %ort]8 ,9 h[ _m po9ts does ! /raight l9e l m t\* ! circumf]5ce8 ,x is call$ a 8tang5t40 ,a tang5t is a /raight l9e t t\*es a circumf]5ce 9 only "o po9t4 ,"n ?ree tang5ts4 ,,lesson ,,?irty-,,?ird4 ,,review4 ,read six *ords4 7,,diagram #bd47 ,:y is i d a *ord8 ,:at is a *ord8 ,"n two diamet]s4 ,:y is a j a diamet]8 ,:at is a diamet]8 ,is e *ord a diamet]8 ,is e diamet] a *ord8 ,"n five radii4 ,:y is c a a radius8 ,:at is a radius8 ,a diamet] is equal to h[ _m radii8 ,are all radii equal to ea* o!r8 ,are all *ords equal to ea* o!r8 ,are all diamet]s equal to ea* o!r8 ,"n two secants4 ,:y is ei!r "o a secant8 ,:at is a secant8 ,"n ?ree tang5ts4 ,:y is a b a tang5t8 ,:at is a tang5t8 ,is a tang5t inside ( a circle or outside ( x8 ,is a *ord inside or outside ( a circle8 ,is a secant )9 or )\t a circle8 ,if ! radius is ?ree in*es1 h[ long is ! diamet]8 ,illu/ration3 ,diagram #be4 ,,lesson ,,?irty-,,f\r?4 ,,arcs ,,& ,,degrees4 ,:at small "p ( ! circumf]5ce ( circle #a 7,,diagram #be47 is m>k$8 ,x is call$ an 8arc40 ,an arc is any "p ( a circumf]5ce4 ,read five arcs t are m>k$4 ,:i* is long]1 ! arc e d1 or ! arc e f8 b d1 or b d e8 a b d1 or a b d e8 ,"n an arc :i* is half ( ! circumf]5ce4 ,x is call$ a 8semi-circumf]5ce40 8,semi0 means 8half40 ,a semi-circumf]5ce is half ( a circumf]5ce4 ,read ?ree arcs1 ea* ( :i* is "o-f\r? ( ! circumf]5ce4 ,if ! :ole circumf]5ce 7 divid$ into ?ree hundr$ & sixty equal arcs1 w\ld ea* arc be l>ge or small8 ,ea* ( ^! arcs w\ld be call$ a 8degree40 ,degrees are m>k$ 774 ,a degree ( a circumf]5ce is a ?ree hundr$ & sixtie? "p ( x4 ,h[ _m degrees 9 a semi-circumf]5ce8 ,h[ _m degrees 9 "o-f\r? ( a circumf]5ce8 ,if a f\r? ( a circumf]5ce 7 divid$ into ?ree equal p>ts1 h[ _m degrees w\ld !re be 9 ea* "p8 ,into h[ _m p>ts w\ld ea* ?ird ( a qu>t] h to be aga9 divid$ to make s+le degrees8 ,is an arc ( n9ety-"o degrees great] or less ?an "o-f\r? ( a circumf]5ce8 ,is an arc ( a hundr$ & sev5ty-n9e degrees great] or less ?an a semi-circumf]5ce8 ,c !re be m ?an ?ree hundr$ & sixty degrees 9 a circumf]5ce8 ,if ! circumf]5ce ( circle #a 7 divid$ into degrees1 ea* degree w\ld be s small an arc t x w\ld look l a dot4 ,if a degree 7 divid$ into sixty equal p>ts1 ea* "p w\ld be call$ a m9ute4 ,if a m9ute 7 divid$ into sixty equal p>ts1 ea* "p w\ld be call$ a second4 ,h[ _m degrees 9 ! l>ge circle ( ,fig4 #b8 ,h[ _m 9 ! small] "o8 ,has a l>ge circle any m degrees ?an a small circle8 ,9 ! l>ge circle h[ _m degrees f a to b8 ,9 ! small circle h[ _m f a to b8 ,:i* is great]1 an arc ( n9ety degrees ( ! l>ge circle1 or "o ( n9ety degrees ( ! small "o8 ,:i* is great]1 an arc ( a degree ( ! l>ge circle1 or "o ( a degree ( ! small "o8 ,! angle a o b has its v]tex at :at "p ( ! l>g] circle8 ,at :at "p ( ! small] circle8 ,on h[ _m degrees ( ! l>g] circle does ! angle /&8 ,on h[ _m degrees ( ! small] circle does x /&8 ,!n x is said to be an angle ( #ij4 ,if ! angle a o f is an angle ( #cj1 h[ _m degrees mu/ !re be 9 ! arc a f8 ,if ! arc f e is an arc ( #fj1 :at is ! size ( ! angle f o e8 ,an angle ( #aj /&s ^u an arc ( h[ _m degrees8 ,( #h8 ,( #a8 ,! angle a o b is :at k9d ( an angle8 ,^u h[ _m degrees does x /&8 ,!n a "r angle is an angle ( h[ _m degrees8 ,if an angle /& ^u less ?an #ij1 :at k9d ( an angle is x8 ,if an angle /& ^u m ?an #ij1 :at k9d ( an angle is x8 ,c an angle h z _m degrees z a hundr$ & eighty8 ,,lesson ,,?irty-,,fif?4 ,,review4 ,read n9e arcs ^: ends are m>k$4 7,,diagram #bf47 ,read ?ree arcs ea* ( :i* is "o-f\r? ( a circumf]5ce4 ,read two arcs ea* ( :i* is "o-half ( a circumf]5ce4 ,:y is e g an arc8 ,:at is an arc8 ,h[ _m degrees 9 ! arc f h8 ,9 e h8 ,if ! arc f h 7 divid$ into ?ree equal p>ts1 h[ _m degrees w\ld !re be 9 ea*8 ,h[ _m degrees 9 a circumf]5ce8 ,9 a semi-circumf]5ce8 ,h[ _m m degrees 9 a l>ge circumf]5ce ?an 9 a small "o8 ,if ! arc i f is #dj1 :at is ! size ( ! angle f o i8 ,if ! angle f o g is an angle ( #acj1 :at is ! size ( ! arc f i h g8 ,h[ _m degrees 9 ea* ( ! adjac5t angles f o h1 h o e8 ,:5 two adjac5t angles are equal to ea* o!r1 :at is ea* call$8 ,h[ _m degrees 9 a "r angle8 ,illu/ration3 ,diagram #bf4 ,,lesson ,,?irty-,,six?4 ,,p>ts ,,( ,,! ,,circle4 ,! "p ( ! circle b\nd$ by ! *ord a b & ! arc a b is call$ a segm5t4 ,read ?ree segm5ts1 ea* less ?an half a circle1 ?us1--! segm5t b\nd$ by ! *ord a d & ! arc a b d4 ,a segm5t is a "p ( a circle b\nd$ by an arc & a *ord4 ,read two segm5ts t are ea* half a circle4 ,:at is ! *ord call$8 ,:at is ! arc call$8 ,a segm5t b\nd$ by a diamet] & a semi-circumf]5ce is a 8semicircle40 ,a semicircle is half a circle4 ,read f\r segm5ts ea* l>g] ?an a semicircle4 ,! "p ( ! circle betwe5 ! two radii o f1 o i1 & ! arc f i1 is call$ a 8sector40 ,read f\r sectors ea* less ?an "o-f\r? ( a circle4#b ,footnote #b3 ,?us1 a sector b\nd$ by ! two radii o g1 o h1 & ! arc g h4 ,a sector is a "p ( a circle b\nd$ by two radii & an arc4 ,:at "p ( ! :ole circle is ! sector f o h8 ,x is call$ a 8quadrant40 ,a quadrant is a sector :i* is "o-f\r? ( a circle4 ,read a sector :i* is great] ?an a quadrant4 ,if ! *ord e f be reg>d$ a diamet]1 :at d y call ! semicircle bel[ x8 ,if x be reg>d$ z two radii1 :at is ! semicircle call$8 ,!n a semicircle is bo? a segm5t & a sector4 ,,lesson ,,?irty-,,sev5?4 ,,review4 ,"n t5 segm5ts4 7,,diagram #bf47 ,:at is a segm5t8 ,( ! segm5ts nam$1 :i* are less ?an a semicircle8 ,:i* are great]8 ,:i* two are semicircles8 ,:i* two are on ! *ord a f8 ,"n n9e sectors4 ,:y is g o i a sector8 ,:at is a sector8 ,:i* f\r ( ! sectors nam$ are ea* less ?an a quadrant8 ,:i* ?ree are quadrants8 ,:i* two are great] ?an a quadrant8 ,:at "p ( ! circle is bo? a segm5t & a sector8 ,h[ _m quadrants 9 a circle8 ,h[ _m semicircles8 ,,"p ,,second4 ,,axioms ,,& ,,!orems4 ,,axioms ,,illu/rat$4 ,,axiom #a4 ,! triangle ,a is equal to ! triangle ,c4 ,! triangle ,b is also equal to ! triangle ,c4 ,:at d y ?9k ( ! two triangles ,a & ,b8 ,:y8 ,illu/ration ,if two ?+s are sep>ately equal to ! same ?+1 !y are equal to ea* o!r4 ,,axiom #b4 ,! squ>e ,a is equal to ! squ>e ,b4 ,to ! rectangle ,c add ! squ>e ,a1 & we h an ,l po9t+ 9 :at direction8 ,to ! same rectangle ,c add ! squ>e ,b1 & we h an ,l po9t+ 9 :at direction8 ,illu/ration ,:i* is l>g]1 ! ,l po9t+ to ! left1 or t po9t+ to ! "r8 ,to :at same ?+ did y add two equals8 ,:at two equals did y add to x8 ,:at 0 ! fir/ sum8 ,! second8 ,:at d y ?9k ( ! two sums8 ,if equals be add$ to ! same ?+1 ! sums w be equal4 ,,axiom #c4 ,illu/ration ,! squ>e ,a is equal to ! squ>e ,b4 ,f ! inv]t$ ,t take away ! squ>e ,a1 & we h an ,l po9t+ 9 :at direction8 ,f ! same ,fig4 ,t take away ! squ>e ,b1 & we h an ,l po9t+ 9 :at direction8 ,:i* is l>g]1 ! ,l po9t+ to ! "r1 or t po9t+ to ! left8 ,:at two equal ?+s did we take away f ! same ?+8 ,f :at same ?+ did we take !m away8 ,:at did we f9d true ( ! two rema9d]s8 ,if equals be tak5 f ! same ?+1 ! rema9d]s w be equal4 ,,axiom #d4 ,illu/ration ,! rectangle #a #b is equal to ! rectangle #a #c4 ,f ! rectangle #a #b take away ! squ>e ,a1 & :at rectangle rema9s8 ,f ! rectangle #a #c take away ! same squ>e ,a1 & :at rectangle rema9s8 ,:i* is great]1 ! rectangle ,b1 or ! rectangle ,c8 ,:at same ?+ did we take away f equals8 ,f :at did we fir/ take x8 ,:at rema9$8 ,f :at did we next take x8 ,:at rema9$8 ,:at did we f9d true ( ! two rema9d]s8 ,if ! same ?+ be tak5 f equals1 ! rema9d]s w be equal4 ,,axiom #e4 ,if equals be add$ to equals1 ! sums w be equal4 ,,axiom #f4 ,if equals be subtract$ f equals1 ! rema9d]s w be equal4 ,,axiom #g4 ,if ! halves ( two ?+s are equal1 ! :oles w be equal4 ,,axiom #h4 ,e ,:ole is equal to ! sum ( all its p>ts4 ,,axiom #i4 ,f "o po9t to ano!r only "o /raight l9e c be drawn4 ,,axiom #aj4 ,a /raight l9e is ! %orte/ di/ance betwe5 two po9ts4 ,,axiom #aa4 ,if two ?+s co9cide ?r\gh\t _! :ole ext5t1 !y are equal4 ,,!orems ,,illu/rat$4 ,illu/ration3 ,diagram #bi4 ,,developm5t ,,lesson4 ,d ! angles ,blue1 ,r$1 take up all ! space on ! l9e a b8 ,d ! angles ,blue1 ,yell[1 ,r$1 take up all ! space on ! l9e8 ,d ! angles ,blue1 ,yell[1 ,gre51 ,r$1 take up all ! space on ! l9e8 ,is !re room betwe5 any two ( ! angles to put 9 ano!r angle8 ,!n are n ! angles ,blue1 ,yell[1 ,gre51 ,r$1 equal to all ! space on ! l9e a b8 ,,note4--,! ^w space1 z "h us$1 means angul> space2 & x is indisp5sable t ! tea*] impress ?is fact ^u ! le>n]4 ,by means ( =m] lessons1 ! pupil has le>n$ positively1 t an angle is ! diff]5ce betwe5 ! directions ( two l9es2 &1 impli$ly1 t ! includ$ space has no?+ to d ) ! size ( ! angle4 ,!re _c1 !re=e1 be mu* dang] t ! pupil w imbibe any errone\s notion f ?is /yle ( expression1 :i* is v mu* m simple ?an to say t ! diff]5ce ( direction ( two giv5 l9es is equal to ! diff]5ce ( direction ( two o!r giv5 l9es1 :i* /yle w be us$ some:at lat] 9 ^! lessons4 ,illu/ration3 ,diagram #cj4 ,,proposition ,i4 ,,!orem4 ,,developm5t ,,lesson4 ,are ! adjac5t angles ,gre51 ,r$1 equal to all ! angul> space on ! l9e a b8 ,place a pap] squ>e corn] or "r angle on ! l9e a b at ! left ( c d ) its v]tex at c4 ,x w cov] all ! angle ,gre5 & "p ( ! angle ,r$ up to ! l9e c d4 ,n[ place ano!r squ>e corn] on ! l9e a b to ! "r ( ! l9e c d1 & ) its v]tex at ! po9t c4 ,x w cov] ! rema9+ "p ( ! angle ,r$1 & two edges ( ! squ>e corn]s w meet along ! l9e c d4 ,are ! two "r angles equal to all ! angul> space on ! l9e a b8 ,!n if ! two adjac5t angles ,gre51 ,r$1 are equal to all ! angul> space on ! l9e a b1 & ! two "r angles are also equal to ! same space1 :at d y inf] conc]n+ ! adjac5t angles & ! two "r angles8 ,:at axiom d y apply :5 y say t ! adjac5t angles are equal to ! two "r angles8 ,to :at same ?+ did y f9d two ?+s sep>ately equal8 ,:at did y fir/ see equal to x8 ,:at did y next see equal to x8 ,!n :at did y f9d true8 ,if ! angle ,r$ 7 small]1 & ! angle ,gre5 l>g]1 w\ld ! adjac5t angles /ill be equal to two "r angles8 ,!n1-- ,any two adjac5t angles are equal to two "r angles4 ,if we draw ! /raight l9e c d :]e ! edges ( ! squ>e corn]s come toge!r1 :at k9d ( angles w a c d1 d c b1 be8 ,see n[ if y c und]/& ! foll[+ demon/ration3-- ,,demon/ration4 ,we wi% to prove t ,any two adjac5t angles are equal to two "r angles4 ,let ! two /raight l9es a b1 m n1 int]sect ea* o!r 9 ! po9t c4 7,,diagram #cj47 ,!n w any two adjac5t angles1 z ,gre51 ,r$1 be equal to two "r angles8 ,=1 f ! po9t c1 draw ! /raight l9e c d s z to make ! angles a c d1 d c b1 "r angles4 ,! adjac5t angles ,gre51 ,r$1 are equal to all ! angul> space on ! l9e a b4 ,! "r angles a c d1 d c b1 are also equal to all ! angul> space on ! l9e a b4 ,!re=e ! adjac5t angles ,gre51 ,r$1 are equal to two "r angles4 ,,te/ ,,que/ions4 ,to :at same ?+ did y f9d two ?+s equal8 ,:at did y fir/ see equal to x8 ,:at did y next see equal to x8 ,!n :at new ?+ did y f9d true8 ,:at axiom did y make use (8 ,illu/ration3 ,diagram #ca4 ,,te/ ,,lesson4 ,by means ( ,fig4 ,a1-- #a4 ,prove t ! adjac5t angles ,gre51 ,r$1 are equal to two "r angles4 #b4 ,prove t ! adjac5t angles ,blue1 ,yell[1 are equal to two "r angles4 ,by means ( ,fig4 ,b1-- #c4 ,prove t ! adjac5t angles ,gre51 ,r$1 are equal to two "r angles4 #d4 ,prove t ! adjac5t angles ,yell[1 ,blue1 are equal to two "r angles4 ,by means ( ,fig4 ,c1-- #e4 ,prove t ! adjac5t angles ,r$1 ,blue1 are equal to two "r angles4 #f4 ,prove t ! adjac5t angles ,gre51 ,yell[1 are equal to two "r angles4 #g4 ,give ! prec$+ demon/rations aga91 b "n ! angles by _! lett]s in/ead ( by _! colors4 ,illu/ration3 ,diagram #cb4 ,,te/ ,,lesson4 ,by means ( ,fig4 ,a prove1-- #a4 ,t ! adjac5t angles a c m1 m c b1 are equal to two "r angles4 #b4 ,t ! adjac5t angles a c n1 n c b1 are equal to two "r angles4 ,by means ( ,fig4 ,b prove1-- #c4 ,t ! adjac5t angles a c n1 n c b1 are equal to two "r angles4 #d4 ,t ! adjac5t angles a c m1 m c b1 are equal to two "r angles4 ,by means ( ,fig4 ,c prove1-- #e4 ,t ! adjac5t angles a c m1 m c b1 are equal to two "r angles4 #f4 ,t ! adjac5t angles a c n1 n c b1 are equal to two "r angles4 ,by means ( ,fig4 ,d prove1-- #g4 ,t ! adjac5t angles a c n1 n c b1 are equal to two "r angles4 #h4 ,t ! adjac5t angles b c m1 m c a1 are equal to two "r angles4 ,illu/ration3 ,diagram #cc4 ,,proposition ,,ii4 ,,!orem4 ,,developm5t ,,lesson4 ,:at k9d ( angles are ,p & ,s8 ,h[ d ! adjac5t angles ,yell[1 ,blue1 comp>e ) ! "r angles ,p1 ,s8 ,h[ d ! adjac5t angles ,blue1 ,r$1 comp>e ) ! two "r angles8 ,!n if ! adjac5t angles ,yell[1 ,blue1 are equal to two "r angles1 & ! adjac5t angles ,blue1 ,r$1 are also equal to two "r angles1 :at d y ?9k ( ! two pairs ( adjac5t angles1 ,yell[1 ,blue1 & ,blue1 ,r$8 ,if1 f ! adjac5t angles ,yell[1 ,blue1 we take away ! angle ,blue1 :at rema9s8 ,if1 f ! adjac5t angles ,blue1 ,r$1 we take away ! same angle ,blue1 :at rema9s8 ,!n1 s9ce ! same angle ,blue has be5 tak5 f equal pairs ( adjac5t angles1 :at d y ?9k ( ! two rema9d]s1 ,yell[1 ,r$8 ,suppose ! l9es a b & m n 7 s drawn t ! angles ,yell[1 ,r$1 7 l>g] or small]1 w\ld !y /ill be equal to ea* o!r8 ,!n1-- ,all v]tical angles are equal to ea* o!r4 ,illu/ration3 ,diagram #cd4 ,,demon/ration4 ,we wi% to prove t ,all v]tical angles are equal to ea* o!r4 ,let ! /raight l9es a b1 m n1 int]sect ea* o!r at ! po9t c1 !n w any two v]tical angles1 z ,yell[1 ,r$1 be equal to ea* o!r4 ,= ! adjac5t angles ,yell[1 ,blue1 are equal to two "r angles4#c ,footnote #c3 ,:5 ?is comp>ison is made1 let ! pupil look at ! "r angles ,p & ,s4 ,! adjac5t angles ,blue1 ,r$1 are also equal to two "r angles4 ,!re=e ! adjac5t angles ,yell[1 ,blue1 are equal to ! adjac5t angles ,blue1 ,r$4 ,if1 f ! adjac5t angles ,yell[1 ,blue1 we take away ! angle ,blue1 we %all h left ! angle ,yell[4 ,if1 f ! adjac5t angles ,blue1 ,r$1 we take away ! same angle ,blue1 we %all h left ! angle ,r$4 ,!re=e ! v]tical angles ,yell[1 ,r$1 are equal to ea* o!r4 ,,te/ ,,que/ions4 ,:5 y say t ! adjac5t angles ,yell[1 ,blue1 are equal to two "r angles1 d y "k x because y see x1 or because y h prov$ x8 ,h[ d y "k t ! adjac5t angles ,blue1 ,r$1 are equal to two "r angles8 ,:5 y say ! adjac5t angles ,yell[1 ,blue1 are equal to ! adjac5t angles ,blue1 ,r$1 :at axiom d y use8 ,:at same ?+ d y take away f equals8 ,f :at equals d y take x away8 ,:5 y take ! angle ,blue f ! adjac5t angles ,yell[1 ,blue1 :at is ! rema9d]8 ,:5 y take ! same angle ,blue f ! adjac5t angles ,blue1 ,r$1 :at is ! rema9d]8 ,:at d y f9d true ( ! two rema9d]s8 ,:at axiom d y use8 ,illu/ration3 ,diagram #ce4 ,,o!r ,,me?ods ,,( ,,demon/ration4 ,! adjac5t angles ,yell[1 ,gre51 are equal to :at8 ,! adjac5t angles ,gre51 ,r$1 are equal to :at8 ,!n :at d y "k ( ! two pairs ( adjac5t angles ,yell[1 ,gre51 & ,gre51 ,r$8 ,f ! adjac5t angles ,yell[1 ,gre51 take away ! angle ,gre54 ,:at rema9s8 ,f ! adjac5t angles ,gre51 ,r$1 take ! same angle ,gre54 ,:at rema9s8 ,:at d y "k ( ! two rema9d]s8 ,:y8 ,:at axiom d y use8 ,9 ! la/ lesson1 :5 y prov$ ! v]tical angles ,yell[1 ,r$1 equal to ea* o!r1 y made use ( ! angle ,blue2 n[ prove ! same two angles equal by means ( ! angle ,gre54 ,! adjac5t angles ,blue1 ,r$1 are equal to :at8 ,! adjac5t angles ,r$1 ,gre51 are equal to :at8 ,!n :at d y "k ( ! two pairs ( adjac5t angles ,blue1 ,r$1 & ,r$1 ,gre58 ,f ! adjac5t angles ,blue1 ,r$1 take away ! angle ,r$4 ,:at rema9s8 ,f ! adjac5t angles ,r$1 ,gre51 take away ! same angle ,r$4 ,:at rema9s8 ,!n :at d y "k ( ! two rema9d]s1 ,blue1 ,gre58 ,n[ apply ! prec$+ demon/ration to ! v]tical angles ,blue1 ,gre54 ,prove ! v]tical angles ,blue1 ,gre51 equal to ea* o!r by means ( ! angle ,yell[4 ,illu/ration3 ,diagram #cf4 ,,te/ ,,lesson4 ,by means ( ,fig4 ,a1-- #a4 ,prove t ! v]tical angles ,yell[1 ,r$1 are equal to ea* o!r1 us+ ! angle ,gre54 #b4 ,prove ! same ?+1 us+ ! angle ,blue4 #c4 ,prove t ! v]tical angles ,blue1 ,gre51 are equal to ea* o!r1 us+ ! angle ,yell[4 #d4 ,prove ! same ?+1 us+ ! angle ,r$4 ,by means ( ,fig4 ,b1-- #e4 ,prove ! v]tical angles ,yell[1 ,r$1 equal to ea* o!r1 us+ ! angle ,gre54 #f4 ,prove ! same ?+1 us+ ! angle ,blue4 #g4 ,prove ! v]tical angles ,gre51 ,blue1 equal by means ( ! angle ,r$4 #h4 ,prove ! same ?+ by means ( ! angle ,yell[4 ,g ?r\gh ! prec$+ eight demon/rations aga91 call+ ! angles by _! lett]s in/ead ( by _! colors4 ,by means ( ,fig4 ,c1 prove t #i4 a c n equals m c b1 by means ( a c m4 #aj4 a c n equals m c b1 by means ( b c n4 #aa4 a c m equals n c b1 by means ( a c n4 #ab4 a c m equals n c b1 by means ( m c b4 ,by means ( ,fig4 ,d1 prove t #ac4 m c a equals b c n1 by means ( a c n4 #ad4 m c a equals b c n1 by means ( m c b4 #ae4 m c b equals a c n1 by means ( m c a4 #af4 m c b equals a c n1 by means ( b c n4 ,illu/ration3 ,diagram #cg4 ,,proposition ,,iii4 ,,!orem4 ,,developm5t ,,lesson4 ,9 ! above diagram1 ! l9es a b1 c d1 are p>allel1 & are int]sect$ by ! l9e e f at ! po9ts m & n4 ,! angle ,r$ measures ! diff]5ce ( direction betwe5 ! l9e m b & :at o!r l9e8 ,! angle ,yell[ measures ! diff]5ce ( direction betwe5 ! l9e n d & :at o!r l9e8 ,!n1 z ! l9es m b & n d are p>allel1 mu/ !re n be ! same diff]5ce ( direction betwe5 !m & ! l9e e f8 ,!n c !re be any diff]5ce betwe5 ! angles :i* measure ^? equal directions8 ,!n :at d y ?9k ( ! opposite ext]ior & int]ior angles ,r$1 ,yell[8 ,,demon/ration4 ,we wi% to prove t ,opposite ext]ior & int]ior angles are equal to ea* o!r4 ,let ! /raight l9e e f int]sect ! two p>allel /raight l9es a b1 c d1 at ! po9ts m & n4 ,!n w any two opposite ext]ior & int]ior angles1 z ,r$1 ,yell[1 be equal to ea* o!r4 ,= ! angle ,r$ measures ! diff]5ce ( direction ( ! l9es m b & e f4 ,& ! angle ,yell[ measures ! diff]5ce ( direction ( ! l9es n d & e f4 ,b because ! l9es m b1 n d1 are p>allel1 ^! diff]5ces are equal4 ,!re=e ! angles :i* measure !m are equal2 t is1 ,! opposite ext]ior & int]ior angles ,r$1 ,yell[1 are equal to ea* o!r4 ,illu/ration3 ,diagram #ch4 ,,te/ ,,lesson4 ,by means ( ,fig4 ,a1-- #a4 ,prove t ! opposite ext]ior & int]ior angles ,gre51 ,blue1 are equal to ea* o!r4 #b4 ,prove t ! opposite ext]ior & int]ior angles ,r$1 ,yell[1 are equal to ea* o!r4 #c4 ,prove ! opposite ext]ior & int]ior angles c n e1 a m n1 equal4 #d4 ,prove ! opposite ext]ior & int]ior angles e n d1 n m b1 equal4 ,by means ( ,fig4 ,b1-- #e4 ,prove ! opposite ext]ior & int]ior angles e m a1 m n d1 equal4 #f4 ,prove ! opposite ext]ior & int]ior angles a m n1 d n f1 equal4 #g4 ,prove ! opposite ext]ior & int]ior angles e m b1 m n c1 equal4 #h4 ,prove ! opposite ext]ior & int]ior angles b m n1 c n f1 equal4 ,illu/ration3 ,diagram #ci4 ,,proposition ,,iv4 ,,!orem4 ,,developm5t ,,lesson4 ,:at d y "k ( ! opposite ext]ior & int]ior angles ,r$1 ,yell[8 ,:at d y "k ( ! v]tical angles ,r$1 ,gre58 ,!n if ! int]ior alt]nate angles ,gre51 ,yell[1 are sep>ately equal to ! angle ,r$1 :at new fact d y "k8 ,:at axiom d y employ8 ,to :at same ?+ did y f9d two ?+s equal8 ,:at two ?+s did y f9d equal to x8 ,,demon/ration4 ,we wi% to prove t ,any two int]ior alt]nate angles are equal to ea* o!r4 ,let ! /raight l9e e f int]sect ! two p>allel /raight l9es a b1 c d1 9 ! po9ts m & n4 ,!n w any two int]ior alt]nate angles1 z ,gre51 ,yell[1 be equal to ea* o!r4 ,= ! opposite ext]ior & int]ior angles ,r$1 ,yell[1 are equal4 ,! v]tical angles ,r$1 ,gre51 are also equal4 ,!n because ! int]ior alt]nate angles ,gre51 ,yell[1 are sep>ately equal to ! angle ,r$1 !y are equal to ea* o!r4 ,illu/ration3 ,diagram #dj4 ,,te/ ,,lesson4 ,:at d y "k ( ! v]tical angles ,gre51 ,r$1 9 ,fig4 ,a8 ,:at d y "k ( ! opposite ext]ior & int]ior angles ,r$1 ,yell[8 ,!n if ! int]ior alt]nate angles ,gre51 ,yell[1 are sep>ately equal to ! angle ,r$1 :at d y inf]8 ,by means ( ,fig4 ,a1-- #a4 ,prove t ! int]ior alt]nate angles ,gre51 ,yell[1 are equal1 us+ ! angle ,r$4 #b4 ,prove ! same angles equal1 us+ ! angle ,blue4 #c4 ,g ?r\gh ! same demon/rations aga91 call+ ! angles by _! lett]s in/ead ( by _! colors4 ,by means ( ,fig4 ,b1-- #d4 ,prove ! int]ior alt]nate angles ,r$1 ,blue1 equal1 us+ ! angle ,yell[4 #e4 ,prove ! same angles equal1 us+ ! angle ,gre54 #f4 ,g ?r\gh ! same two demon/rations aga91 nam+ ! angles by _! lett]s in/ead ( by _! colors4 ,by means ( ,fig4 ,c1-- #g4 ,prove ! int]ior alt]nate angles c n m1 n m b1 equal1 us+ ! angle f n d4 #h4 ,prove ! same1 us+ ! angle a m e4 #i4 ,prove ! int]ior alt]nate angles a m n1 m n d1 equal1 us+ ! angle e m b4 #aj4 ,prove ! same1 us+ ! angle c n f4 ,illu/ration3 ,diagram #da4 ,,proposition ,v4 ,,!orem4 ,,developm5t ,,lesson4 ,:at d y "k ( ! opposite ext]ior & int]ior angles ,r$1 ,yell[8 ,:at d y "k ( ! v]tical angles ,yell[1 ,gre58 ,!n if ! ext]ior alt]nate angles ,r$1 ,gre51 are sep>ately equal to ! angle ,yell[1 :at new ?+ d y "k to be true8 ,:at axiom d y employ8 ,to :at same ?+ did y "k two ?+s to be equal8 ,:at two ?+s did y "k to be equal to x8 ,!n :at new ?+ did y f9d to be true8 ,,demon/ration4 ,we wi% to prove t ,any two ext]ior alt]nate angles are equal to ea* o!r4 ,let ! /raight l9e e f int]sect ! two p>allel /raight l9es a b1 c d1 at ! po9ts m & n4 ,!n w any two ext]ior alt]nate angles1 z ,r$1 ,gre51 be equal4 ,= ! opposite ext]ior & int]ior angles ,r$1 ,yell[1 are equal to ea* o!r4 ,& ! v]tical angles ,yell[1 ,gre51 are also equal to ea* o!r4 ,!n because ! ext]ior alt]nate angles ,r$1 ,gre51 are sep>ately equal to ! angle ,yell[1 !y are equal to ea* o!r4 ,illu/ration3 ,diagram #db4 ,,te/ ,,lesson4 ,:at d y "k ( ! opposite ext]ior & int]ior angles ,yell[1 ,r$8 ,:at d y "k ( ! v]tical angles ,r$1 ,blue8 ,!n if ! ext]ior alt]nate angles ,yell[1 ,blue1 are sep>ately equal to ! angle ,r$1 :at d y "k ( !m8 ,by means ( ,fig4 ,a1-- #a4 ,prove t ! ext]ior alt]nate angles ,yell[1 ,blue1 are equal1 us+ ! angle ,r$4 #b4 ,prove ! same ?+1 us+ ! angle ,gre54 #c4 ,g ?r\gh ! same demon/rations1 call+ ! angles by _! lett]s4 #d4 ,prove ! ext]ior alt]nate angles e m b1 c n f1 equal1 us+ ! angle a m n4 #e4 ,prove ! same1 us+ ! angle m n d4 ,by means ( ,fig4 ,b1-- #f4 ,prove t ! ext]ior alt]nate angles c m e1 f n b1 are equal1 us+ ! angle n m d4 #g4 ,prove ! same1 us+ ! angle a n m4 #h4 ,prove ! ext]ior alt]nate angles e m d1 a n f1 equal1 us+ ! angle c m n4 #i4 ,prove ! same1 us+ ! angle m n b4 ,illu/ration3 ,diagram #dc4 ,,proposition ,,vi4 ,,!orem4 ,,developm5t ,,lesson4 ,:at d y "k ( ! int]ior alt]nate angles ,yell[1 ,r$8 ,if to ! angle ,gre5 y add ! angle ,yell[1 :at is ! sum8 ,if to ! same angle ,gre5 y add ! equal angle ,r$1 :at is ! sum8 ,!n1 hav+ add$ equals to ! same ?+1 :at d y ?9k ( ! two sums1--! adjac5t angles ,gre51 ,yell[1 & ! int]ior opposite angles ,gre51 ,r$8 ,:at d y "k ( ! adjac5t angles ,gre51 ,yell[1 & ! "r angles ,p1 ,s8 ,!n if ! int]ior opposite angles ,gre51 ,r$1 & ! two "r angles ,p1 ,s1 are sep>ately equal to ! adjac5t angles ,gre51 ,yell[1 :at new ?+ d y "k8 ,,demon/ration4 ,we wi% to prove t ,any two int]ior opposite angles are equal to two "r angles4 ,let ! /raight l9e e f int]sect ! two p>allel /raight l9es a b1 c d1 9 ! po9ts m & n4 ,!n w any two int]ior opposite angles be equal to two "r angles4 ,= ! int]ior alt]nate angles ,yell[1 ,r$1 are equal4 ,if to ! angle ,gre5 we add ! angle ,yell[1 we %all h ! adjac5t angles ,gre51 ,yell[4 ,if to ! same angle ,gre5 we add ! equal angle ,r$1 we %all h ! int]ior opposite angles ,gre51 ,r$4 ,!n ! adjac5t angles ,gre51 ,yell[1 are equal to ! int]ior opposite angles ,gre51 ,r$4 ,b ! adjac5t angles ,gre51 ,yell[1 are equal to two "r angles4 ,!n because ! int]ior opposite angles ,gre51 ,r$1 & two "r angles1 are sep>ately equal to ! two adjac5t angles ,gre51 ,yell[1 !y are equal to ea* o!r4 ,illu/ration3 ,diagram #dd4 ,,te/ ,,lesson4 ,by means ( ,fig4 ,a1-- #a4 ,prove ! int]ior opposite angles ,gre51 ,yell[1 equal to two "r angles1 us+ ! angle ,r$4 #b4 ,prove ! same1 us+ ! angle ,blue4 #c4 ,prove ! same1 us+ ! angle e g b4 #d4 ,prove ! same1 us+ ! angle f h d4 #e4 ,g ?r\gh ! same demon/rations aga91 nam+ ! angles by _! lett]s in/ead ( by _! colors4 #f4 ,prove ! int]ior opposite angles ,r$1 ,blue1 equal to two "r angles1 us+ ! angle ,yell[4 #g4 ,prove ! same1 us+ ! angle ,gre54 #h4 ,prove ! same1 us+ ! angle e g a4 #i4 ,prove ! same1 us+ ! angle c h f4 #aj4 ,g ?r\gh ! same demon/rations aga91 call+ ! angles by _! lett]s in/ead ( by _! colors4 ,by means ( ,fig4 ,b1-- #aa4 ,prove ! int]ior opposite angles a g h1 g h c1 equal to two "r angles1 us+ ! angle g h d4 #ab4 ,prove ! same1 us+ ! angle c h f4 #ac4 ,prove ! same1 us+ ! angle a g e4 #ad4 ,prove ! int]ior opposite angles b g h1 g h d1 equal to two "r angles1 us+ ! angle a g h4 #ae4 ,prove ! same1 us+ ! angle e g b4 #af4 ,prove ! same1 us+ ! angle f h d4 ,comp>e ! angles ,yell[1 ,gre51 ea* ) its ext]ior opposite angle1 & see if y c prove t ! ext]ior opposite angles e g b1 f h d1 are also equal to two "r angles4 ,illu/ration3 ,diagram #de4 ,,proposition ,,vii4 ,,!orem4 ,,developm5t ,,lesson4 ,suppose we d n "k :e!r ! l9es a b1 c d1 are p>allel1 or n2 ,b1 by measur+1 we f9d t ! int]ior angles ,blue1 ,yell[1 on ! same side ( ! secant#d l9e e f1 are equal to two "r angles3 ,footnote #d3 8,secant0 means 8cutt+40 ,! adjac5t angles ,blue1 ,r$1 are equal to :at8 ,!n1 if ! int]ior angles ,blue1 ,yell[1 are equal to two "r angles1 ,& ! adjac5t angles ,blue1 ,r$1 are also equal to two "r angles1 ,:at d y inf]8 ,f ! int]ior angles ,blue1 ,yell[1 take away ! angle ,blue3 :at rema9s8 ,f ! adjac5t angles ,blue1 ,r$1 take away ! same angle ,blue3 :at rema9s8 ,:at d y "k ( ! two rema9d]s8 ,! angle ,r$ measures ! direction ( ! l9e g b f :at l9e8 ,! equal angle ,yell[ measures ! direction ( ! l9e h d f :at l9e8 ,!n if ! l9es g b1 h d1 h ! same direction f ! l9e e f1 :at d y call !m8 ,illu/ration3 ,diagram #df4 ,,demon/ration4 ,we wi% to prove1 t1 ,if a /raight l9e int]sects two o!r /raight l9es s t two int]ior angles on ! same side ( ! int]sect+ l9e are equal to two "r angles1 ! two l9es are p>allel4 ,let ! /raight l9e e f int]sect ! two /raight l9es a b1 c d1 9 ! po9ts g & h1 s t ! angles ,r$1 ,blue1 are equal to two "r angles4 ,!n w ! l9es a b1 c d1 be p>allel4 ,= ! angles ,r$1 ,blue1 are suppos$ equal to two "r angles4 ,! adjac5t angles ,r$1 ,gre51 are kn[n to be also equal to two "r angles4 ,!n ! int]ior angles ,r$1 ,blue1 are equal to ! adjac5t angles ,r$1 ,gre54 ,if f ! int]ior angles ,r$1 ,blue1 we take away ! angle ,r$1 we h left ! angle ,blue4 ,if f ! adjac5t angles ,r$1 ,gre51 we take ! same angle ,r$1 we %all h left ! angle ,gre54 ,!n ! angle ,blue is equal to ! angle ,gre54 ,b ! angle ,blue measures ! direction ( ! l9e h d f ! l9e e f4 ,& ! angle ,gre5 measures ! direction ( ! l9e g b f ! l9e e f4 ,!n ! l9es g b1 h d1 h ! same direction1 & are p>allel4 ,,te/ ,,lesson4 #a4 ,prove ! same )\t ! colors4 #b4 ,prove ! same1 us+ ! angle f h d4 #c4 ,prove ! same1 suppos+ ! angles a g h1 g h c1 equal to two "r angles1 & us+ ! angle a g e4 #d4 ,prove ! same1 us+ ! angle c h f4 ,see ,note ,e1 ,app5dix4 ,,proposition ,,viii4 ,,!orem4 ,! foll[+ demon/ration is v easy4 ,read x once1 & see if y c g ?r\gh x )\t a second read+3-- ,,demon/ration4 ,illu/ration ,we wi% to prove t ,! sum ( any two sides ( a triangle is great] ?an ! ?ird side4 ,let ! figure a b c be a triangle1 !n w ! sum ( any two sides1 z a c1 c b1 be great] ?an ! ?ird side a b4 ,= ! /raight l9e a b is ! %orte/ di/ance betwe5 ! two po9ts a & b1 & is !re=e less ?an ! brok5 l9e a c b4 ,,proposition ,,ix4 ,,problem4 ,! foll[+ solution is s easy t y w und]/& x at once3-- ,we wi% ,to con/ruct an equilat]al triangle on a giv5 /raight l9e4 ,illu/ration ,,solution4 ,let a b be ! giv5 l9e4 ,) ! po9t a z a c5tre1 & a b z a radius1 draw ! circumf]5ce ( ! circle1 or a "p ( "o4 ,) ! po9t b z a c5tre1 & ! same radius a b1 draw ano!r circumf]5ce1 or a "p ( "o4 ,f ! po9t c1 9 :i* ! circumf]5ces or arcs int]sect1 draw ! /raight l9es a c & b c4 ,n[1 because ! l9es a b & a c are radii ( ! same circle1 !y are equal4 ,&1 because ! l9es a b & b c are radii ( ! same circle1 !y are also equal4 ,!n1 because ! two l9es a c1 b c1 are sep>ately equal to ! l9e a b1 !y are equal to ea* o!r1 & ! triangle is equilat]al4 ,illu/ration ,,proposition ,x4 ,,!orem4 ,,developm5t ,,lesson4 ,let ! figure a b c be a triangle4 ,produce ! side a c to d4 ,we h n[ ano!r angle1 b c d1 & we wi% to f9d out if x is equal to any ( ! angles ( ! triangle4 ,f ! po9t c draw ! l9e c e p>allel to a b4 ,because ! /raight l9e a d int]sects ! two p>allels a b1 c e1 ! angle a is equal to :at o!r angle8 ,because ! /raight l9e b c int]sects ! two p>allels a b1 c e1 ! angle b is equal to :at o!r angle8 ,!n ! angles a & b are equal to :at two angles8 ,h[ does ! angle b c d comp>e ) ! angles b c e1 e c d8 ,!n1 if ! angles a & b1 on ! "o h&1 & ! angle b c d1 on ! o!r1 are sep>ately equal to ! angles b c e1 e c d1 ,:at h y f\nd out8 ,:at axiom h y j employ$8 ,to :at same ?+ h y f\nd two o!r ?+s equal8 ,:at two ?+s did y f9d equal to x8 ,,demon/ration4 ,we wi% to prove1 t1 ,if any side ( a triangle be produc$1 ! new angle =m$ w be equal to ! sum ( ! angles t are n adjac5t to x4 ,let a b c be a triangle4 ,produce ! side a c to d2 !n w ! new angle b c d be equal to ! sum ( ! angles a & b4 ,= f ! po9t c draw c e p>allel to a b4 ,!n1 because ! /raight l9e a d int]sects ! two p>allels a b1 c e1 9 ! po9ts a & c1 ,! opposite ext]ior & int]ior angles a & e c d are equal to ea* o!r4 ,& because ! /raight l9e b c int]sects ! same p>allels 9 ! po9ts b & c1 ,! int]ior alt]nate angles b & b c e are equal4 ,!n ! angles a & b ( ! triangle are equal to ! angles b c e & e c d4 ,b ! new angle b c d is equal to ! angles b c e1 e c d4 ,!n because ! new angle b c d1 & ! angles a & b are sep>ately equal to ! angles b c e1 e c d1 !y are equal to ea* o!r4 ,illu/ration ,,proposition ,,xi4 ,,!orem4 ,,developm5t ,,lesson4 ,let ! figure a b c be a triangle4 ,produce ! side a c to d4 ,by ! la/ !orem1 ! angle b c d is equal to :at angles ( ! triangle8 ,:at angle mu/ we add to ^! angles to make up ! ?ree angles ( ! triangle8 ,if we add ! same angle to ! angle b c d1 :at adjac5t angles d we get8 ,!n ! ?ree angles ( ! triangle1 a1 b1 & c1 are equal to :at two angles8 ,b ! adjac5t angles a c b & b c d are equal to :at8 ,!n1 because ! ?ree angles ( ! triangle1 a1 b1 & c1 & two "r angles1 are sep>ately equal to ! two adjac5t angles c & b c d4 ,:at new ?+ h y f\nd out8 ,,demon/ration4 ,we wi% to prove t ,! ?ree angles ( any triangle are equal to two "r angles4 ,let ! figure a b c be a triangle2 !n w ! sum ( ! angles a1 b1 & c1 be equal to two "r angles4 ,=1 produce ! side a c to d4 ,! new angle b c d is equal to ! sum ( ! angles a & b4 ,if to ! angles a & b we add ! angle c1 we %all h ! ?ree angles ( ! triangle4 ,if to ! angle b c d we add ! same angle c1 we %all h ! adjac5t angles c & b c d4 ,!n ! ?ree angles ( ! triangle a1 b1 c1 are equal to ! adjac5t angles c & b c d4 ,b ! adjac5t angles c & b c d are equal to two "r angles4 ,!n1 because ! ?ree angles ( ! triangle are equal to ! adjac5t angles c & b c d1 !y are equal to two "r angles4 ,illu/ration ,,proposition ,,xii4 ,,!orem4 ,,developm5t ,,lesson4 ,let ! ,fig4 ,a ,b ,c ,d be a p>allelogram4 ,produce ! side ,c ,d to ,f4 ,because ! /raight l9e ,b ,d int]sects ! p>allels ,a ,b & ,c ,f1 ! angle ,b is equal to :at o!r angle8 ,because ! /raight l9e ,c ,f int]sects ! p>allels ,a ,c & ,b ,d1 ! angle ,c is equal to :at o!r angle8 ,!n :at foll[s f ?is8 ,to :at angle did y f9d two o!rs equal8 ,:at two angles did y f9d equal to x8 ,:at axiom d y ?9k (8 ,see if y c g ?r\gh ! demon/ration )\t read+ x ev5 once4 ,,demon/ration4 ,we wi% to prove t ,! opposite angles ( a p>allelogram are equal to ea* o!r4 ,let ! ,fig4 ,a ,b ,c ,d be a p>allelogram4 ,!n w any two opposite angles1 z ,b & ,c1 be equal to ea* o!r4 ,= produce ! l9e ,c ,d to ,f4 ,because ! /raight l9e ,b ,d meets ! two p>allels ,a ,b & ,c ,f1 ,! int]ior alt]nate angles ,b & ,e are equal to ea* o!r4 ,because ! /raight l9e ,c ,f meets ! two p>allels ,b ,d & ,a ,c1 ,! opposite ext]ior & int]ior angles ,c & ,e are equal to ea* o!r4 ,!n1 because ! angles ,b & ,c are sep>ately equal to ! angle ,e1 !y are equal to ea* o!r4 99 99 99 99 99 #a4 ,prove ! same by produc+ ! l9e ,a ,b t[>ds ! left4 #b4 ,prove ! same by produc+ ! l9e ,b ,d d[nw>ds4 #c4 ,prove ! angles ,a & ,d equal to ea* o!r by produc+ ! l9e ,c ,d t[>ds ! left4 #d4 ,prove ! same by produc+ ! l9e ,d ,b upw>ds4 #e4 ,see if y c prove ! same by draw+ a diagonal ?r\gh ! po9ts ,a & ,d4 ,illu/ration ,,proposition ,,xiii4 ,,!orem4 ,,developm5t ,,lesson4 ,9 ^! two triangles we h tri$ to make ! side a b ( ! "o equal to ! side d e ( ! o!r2 ! side a c ( ! "o equal to ! side d f ( ! o!r2 & ! includ$ angle b a c ( ! "o equal to ! includ$ angle e d f ( ! o!r4 ,we n[ wi% to f9d out if ! ?ird side b c ( ! "o is equal to ! ?ird side e f ( ! o!r1 & if ! two rema9+ angles b & c ( ! "o are equal to ! two rema9+ angles e & f ( ! o!r4 ,suppose we 7 to cut ! triangle d e f out ( ! page1 & place x ^u ! triangle a b c1 s t ! l9e d e %\ld fall ^u ! l9e a b1 & ! po9t d ^u ! po9t a4 ,z ! l9e d e is equal to ! l9e a b1 ^u :at po9t w ! po9t e fall8 ,if ! angle e d f 7 less ?an ! angle b a c1 w\ld ! l9e d f fall )9 or )\t ! triangle8 ,if ! angle e d f 7 great] ?an ! angle b a c1 :]e w\ld ! l9e d f fall8 ,s9ce ! angle a is equal to d1 :]e1 !n1 mu/ ! l9e d f fall8 ,z ! l9e d f is equal to ! l9e a c1 ^u :at po9t w ! po9t f fall8 ,!n1 if ! po9t e falls ^u ! po9t b1 & ! po9t f ^u ! po9t c1 :]e w ! l9e e f fall8 ,n[1 because ! ?ree sides ( ! triangle d e f exactly fall ^u ! ?ree sides ( ! triangle a b c1 we say ! two magnitudes co9cide ?r\gh\t _! :ole ext5t1 & are !re=e equal4 ,:at ?ree p>ts ( ! triangle a b c did we suppose to be equal to ?ree correspond+ p>ts ( ! triangle d e f be=e we plac$ "o ^u ! o!r4 ,:at l9e ( ! "o d we f9d equal to a l9e 9 ! o!r8 ,:at two angles ( ! "o d we f9d equal to two angles 9 ! o!r8 ,:at d y ?9k ( ! areas ( ! triangles8 ,illu/ration ,,demon/ration4 ,we wi% to prove1 t1 ,if two triangles h two sides1 & ! includ$ angle ( ! "o equal to two sides & ! includ$ angle ( ! o!r1 ea* to ea*1 ! two triangles are equal 9 all respects4 ,let ! triangles a b c & d e f h ! side a b ( ! "o equal to ! side d e ( ! o!r2 ! side a c ( ! "o equal to ! side d f ( ! o!r2 & ! includ$ angle b a c ( ! "o equal to ! includ$ angle e d f ( ! o!r1 ea* to ea*2 !n w ! two triangles be equal 9 all _! p>ts4 ,=1 place ! triangle d e f ^u ! triangle a b c1 s t ! l9e d e %all fall ^u ! l9e a b1 ) ! po9t d ^u ! po9t a4 ,because ! l9e d e is equal to ! l9e a b1 ! po9t e w fall ^u ! po9t b4 ,because ! angle e d f is equal to ! angle b a c1 ! l9e d f w fall ^u ! l9e a c4 ,because ! l9e d f is equal to ! l9e a c1 ! po9t f w fall ^u ! po9t c4 ,!n1 because ! po9t e is on ! po9t b1 & ! po9t f on ! po9t c1 ! l9e e f w co9cide ) ! l9e b c1 & ! two triangles w be f\nd equal 9 all _! p>ts2 ,t is1 ! angle e is f\nd to be equal to ! angle b1 ! angle f to ! angle c1 ! l9e e f to ! l9e b c1 & ! area ( ! triangle a b c to ! area ( ! triangle d e f4 ,illu/ration ,,proposition ,,xiv4 ,,!orem4 ,,developm5t ,,lesson4 ,9 ^! two triangles we h tri$ to make ! angle b ( ! "o equal to ! angle e ( ! o!r2 ! angle c ( ! "o equal to ! angle f ( ! o!r2 & ! includ$ side b c ( ! "o equal to ! includ$ side e f ( ! o!r4 ,we n[ wi% to f9d out if ! rema9+ angle a ( ! "o is equal to ! rema9+ angle d ( ! o!r1 & if ! two rema9+ sides a b & a c ( ! "o are equal to ! two rema9+ sides d e & d f ( ! o!r4 ,suppose we 7 to cut ! triangle d e f out ( ! page & place x ^u ! triangle a b c1 s t ! l9e e f %all fall ^u ! l9e b c1 ) ! po9t e ^u ! po9t b4 ,because ! l9e e f is equal to ! l9e b c1 ^u :at po9t w ! po9t f fall8 ,because ! angle e is equal to ! angle b1 :]e w ! l9e e d fall8 ,because ! angle f is equal to ! angle c1 :]e w ! l9e d f fall8 ,!n1 if ! l9e d e falls ^u ! l9e a b & ! l9e d f ^u ! l9e a c1 :]e w ! po9t d fall8 ,n[ because ! ?ree sides ( ! triangle d e f exactly fall ^u ! ?ree sides ( ! triangle a b c1 we say ! two magnitudes co9cide ?r\gh\t _! :ole ext5t1 & are !re=e equal4 ,suppose ! angle e 7 great] ?an ! angle b1 w\ld ! l9e e d fall )9 or )\t ! triangle8 ,if x 7 less1 :]e w\ld ! l9e fall8 ,:y does ! l9e d e fall exactly ^u ! l9e a b8 ,illu/ration ,,demon/ration4 ,we wi% to prove t1 ,if two triangles h two angles1 & ! includ$ side ( ! "o equal to two angles & ! includ$ side ( ! o!r1 ea* to ea*1 ! two triangles are equal to ea* o!r 9 all respects4 ,let ! triangles a b c & d e f h ! angle b ( ! "o equal to ! angle e ( ! o!r2 ! angle c ( ! "o equal to ! angle f ( ! o!r2 & ! includ$ side b c ( ! "o equal to ! includ$ side e f ( ! o!r1 ea* to ea*2 !n w ! two triangles be equal 9 all _! p>ts4 ,= place ! triangle d e f ^u ! triangle a b c1 s t ! l9e e f %all fall ^u ! l9e b c1 ) ! po9t e ^u ! po9t b4 ,because ! l9e e f is equal to ! l9e b c ! po9t f w fall ^u ! po9t c4 ,because ! angle e is equal to ! angle b1 ! l9e e d w fall ^u ! l9e b a1 & ! po9t d w be some:]e 9 ! l9e b a4 ,because ! angle f is equal to ! angle c1 ! l9e f d w fall ^u ! l9e c a1 & ! po9t d w be some:]e 9 ! l9e c a4 ,!n1 because ! po9t d is 9 ! two l9es1 b a & c a1 x mu/ be 9 _! int]section1 or ^u ! po9t a4 ,&1 z ! two triangles co9cide ?r\gh\t _! :ole ext5t1 !y are equal 9 all _! p>ts4 ,t is1 ! angle a is f\nd to be equal to ! angle d2 ! side b a to ! side e d2 ! side c a to ! side f d2 & ! area ( ! triangle a b c to ! area ( ! triangle d e f4 ,illu/ration ,,proposition ,,xv4 ,,!orem4 ,,demon/ration4 ,we wi% to prove t ,! opposite sides ( any p>allelogram are equal4 ,let ! figure a b c d be a p>allelogram2 !n w ! sides a b & c d be equal to ea* o!r2 likewise ! sides a d & b c4 ,=1 draw ! diagonal b d4 ,because ! figure is a p>allelogram1 ! sides a b & d c are p>allel1 & ! int]ior alt]nate angles n & o are equal4 ,because ! figure is a p>allelogram1 ! int]ior alt]nate angles r & m are equal4 ,!n ! two triangles a d b1 b d c1 h two angles & ! includ$ side ( ! "o equal to two angles & ! includ$ side ( ! o!r1 ea* to ea*1 & are !re=e equal2 ,& ! side a b opposite ! angle m is equal to ! side c d opposite ! equal angle r2 ,& ! side a d opposite ! angle n is equal to ! side b c opposite ! equal angle o4 ,,te/4 ,prove ! same by draw+ a diagonal f a to c4 ,illu/ration ,,proposition ,,xvi4 ,,!orem4 ,,developm5t ,,lesson4 ,suppose ,a ,b to be a /raight l9e1 & ,c any po9t out ( x4 ,f ! po9t ,c draw a p]p5dicul> ,c ,f to ,a ,b4 ,let u see if ?is p]p5dicul> is n %ort] ?an any o!r l9e we c draw f ! same po9t to ! same l9e4 ,draw any o!r l9e f ,c to ,a ,b z ,c ,e4 ,n[1 z ,c ,e is any l9e :atev] o!r ?an a p]p5dicul>1 if we f9d t ! p]p5dicul> ,c ,f is %ort] ?an x we mu/ conclude t x is ! %orte/ l9e t c be drawn f ,c to ,a ,b4 ,produce ,c ,f until ,f ,d is equal to ,c ,f1 & !n jo9 ,e & ,d4 ,9 ! triangles ,e ,f ,c1 ,e ,f ,d1 :at two sides 7 drawn equal8 ,:at l9e is a side to ea*8 ,h[ great an angle is ,c ,f ,e8 ,:at is a "r angle8 ,!n h[ d ! angles ,c ,f ,e & ,e ,f ,d comp>e ) ea* o!r8 ,if ! two triangles ,e ,f ,c1 ,e ,f ,d1 h ! side ,c ,f ( ! "o equal to ! side ,f ,d ( ! o!r1 ! side ,e ,f common to bo?1 & ! includ$ angle ,e ,f ,c ( ! "o equal to ! includ$ angle ,e ,f ,d ( ! o!r1 ea* to ea*1 :at d y inf]8 ,!n :at ?ird side ( ! "o h y f\nd equal to a ?ird side ( ! o!r8 ,c ,e is :at "p ( ! brok5 l9e ,c ,e ,d8 ,c ,f is :at "p ( ! l9e ,c ,d8 ,:i* is %ort]1 ! /raight l9e ,c ,d1 or ! brok5 l9e ,c ,e ,d8 ,!n h[ does ! half ( ,c ,d or ,c ,f comp>e ) ! half ( ,c ,e ,d or ,c ,e8 ,if ,c ,e is any l9e :atev] o!r ?an a p]p5dicul>1 :at may we n[ say ( ! p]p5dicul> f ! po9t ,c to ! /raight l9e ,a ,b8 ,illu/ration ,,demon/ration4 ,we wi% to prove t ,a p]p5dicul> is ! %orte/ di/ance f a po9t to a /raight l9e4 ,let ,a ,b be a /raight l9e1 & ,c ,a po9t out ( x2 !n w ! p]p5dicul> ,c ,e be ! %orte/ l9e t c be drawn f ! po9t to ! l9e4 ,= draw any o!r l9e f ,c to ,a ,b1 z ,c ,f4 ,produce ,c ,e until ,e ,d equals ,c ,e1 & jo9 ,f ,d4 ,! two triangles ,f ,e ,c1 ,f ,e ,d1 h ! side ,c ,e ( ! "o equal to ! side ,e ,d ( ! o!r1 ! side ,f ,e common1 & ! includ$ angle ,f ,e ,c ( ! "o equal to ! includ$ angle ,f ,e ,d ( ! o!r1 !y are !re=e equal1 & ! side ,c ,f equals ! side ,f ,d4 ,b ! /raight l9e ,c ,d is ! %orte/ di/ance betwe5 ! two po9ts ,c ,d2 !re=e x is %ort] ?an ! brok5 l9e ,c ,f ,d4 ,!n ,c ,e1 ! half ( ,c ,d1 is %ort] ?an ,c ,f1 ! half ,c ,f ,d4 ,&1 z ,c ,f is any l9e o!r ?an a p]p5dicul>1 ! p]p5dicul> ,c ,e is ! %orte/ l9e t c be drawn f ,c to ,a ,b4 ,illu/ration ,,proposition ,,xvii4 ,,!orem4 ,,demon/ration4 ,we wi% to prove t ,a tang5t to a circumf]5ce is p]p5dicul> to a radius at ! po9t ( contact4 ,let ! /raight l9e ,a ,b be tang5t at ! po9t ,d to ! circumf]5ce ( ! circle ^: c5tre is ,c4 ,jo9 ! c5tre ,c ) ! po9t ( contact ,d1 ! tang5t w be p]p5dicul> to ! radius ,c ,d4 ,= draw any o!r l9e f ! c5tre to ! tang5t1 z ,c ,f4 ,z ! po9t ,d is ! only "o 9 :i* ! tang5t t\*es ! circumf]5ce1 any o!r po9t1 z ,f1 mu/ be )\t ! circumf]5ce4 ,!n ! l9e ,c ,f1 rea*+ beyond ! circumf]5ce1 mu/ be long] ?an ! radius ,c ,d1 :i* w\ld rea* only to x2 !re=e ,c ,d is %ort] ?an any o!r l9e :i* c be drawn f ! po9t ,c to ! /raight l9e ,a ,b2 !re=e x is p]p5dicul> to x4 ,,proposition ,,xviii4 ,,!orem4 ,,demon/ration4 ,illu/ration ,we wi% to prove t ,9 any isosceles triangle1 ! angles opposite ! equal sides are equal4 ,let ! triangle ,a ,b ,c be isosceles1 hav+ ! side ,a ,b equal to ! side ,a ,c2 !n w ! angle ,b1 opposite ! side ,a ,c1 be equal to ! angle ,c1 opposite ! equal side ,a ,b4 ,= draw ! l9e ,a ,d s z to divide ! angle ,a into two equal p>ts1 & let x be long en\gh to divide ! side ,b ,c at "s po9t z ,d4 ,n[ ! two triangles ,a ,d ,b1 ,a ,d ,c1 h ! side ,a ,b ( ! "o equal to ! side ,a ,c ( ! o!r1 ! side ,a ,d common to bo?1 & ! includ$ angle ,b ,a ,d ( ! "o equal to ! includ$ angle ,c ,a ,d ( ! o!r2 !re=e ! two triangles are equal 9 all respects1 & ! angle ,b1 opposite ! side ,a ,c1 is equal to ! angle ,c1 opposite ! side ,a ,b4 ,illu/ration ,,proposition ,,xix4 ,,!orem4 ,,demon/ration4 ,we wi% to prove t1 ,if two triangles h ! ?ree sides ( ! "o equal to ! ?ree sides ( ! o!r1 ea* to ea*1 !y are equal 9 all _! p>ts4 ,let ! two triangles ,a ,b ,c1 ,a ,d ,c1 h ! side ,a ,b ( ! "o equal to ! side ,a ,d ( ! o!r2 ! side ,b ,c ( ! "o equal to ! side ,d ,c ( ! o!r1 & ! ?ird side likewise equal2 !n w ! two triangles be equal 9 all _! p>ts4 ,= place ! two triangles toge!r by _! longe/ side1 & jo9 ! opposite v]tices ,b & ,d by a /raight l9e4 ,because ! side ,a ,b is equal to ! side ,a ,d1 ! triangle ,b ,a ,d is isosceles1 & ! angles ,a ,b ,d1 ,a ,d ,b1 opposite ! equal sides are equal4 ,because ! side ,b ,c is equal to ! side ,d ,c1 ! triangle ,b ,c ,d is isosceles1 & ! angles ,c ,b ,d1 ,c ,d ,b1 opposite ! equal sides are equal4 ,if to ! angle ,a ,b ,d we add ! angle ,d ,b ,c1 we %all h ! angle ,a ,b ,c4 ,& if to ! equal ( ,a ,b ,d1 t is1 ,a ,b ,d1 we add ! equal ( ,d ,b ,c1 t is1 ,b ,d ,c1 we %all h ! angle ,a ,d ,c4 ,!re=e ! angle ,a ,b ,c is equal to ! angle ,a ,d ,c4 ,!n ! two triangles ,a ,b ,c1 ,a ,d ,c1 h two sides1 & ! includ$ angle ( ! "o equal to two sides & ! includ$ angle ( ! o!r1 ea* to ea*1 & are equal 9 all _! p>ts2 t is1 ! ?ree angles ( ! "o are equal to ! ?ree angles ( ! o!r1 & _! areas are equal4 ,illu/ration ,,proposition ,,xx4 ,,!orem4 ,,demon/ration4 ,we wi% to prove t ,an angle at ! circumf]5ce is measur$ by half ! arc on :i* x /&s4 ,let ,b ,a ,d be an angle ^: v]tex is 9 ! circumf]5ce ( ! circle ^: c5tre is ,c2 !n w x be measur$ by half ! arc ,b ,d4 ,= ?r\gh ! c5tre draw ! diamet] ,a ,e1 & jo9 ! po9ts ,c & ,b4 ,! ext]ior angle ,e ,c ,b is equal to ! sum ( ! angles ,b & ,b ,a ,c4 ,because ! sides ,c ,a1 ,c ,b1 are radii ( ! circle1 !y are equal1 ! triangle is isosceles1 ! angles ,b & ,b ,a ,c opposite ! equal sides are equal1 & ! angle ,b ,a ,c is half ( bo?4 ,!n1 because ! angle ,b ,a ,c is half ( ,b & ,b ,a ,c1 x mu/ be half ( _! equal ,e ,c ,b4 ,b ,e ,c ,b1 be+ at ! c5tre1 is measur$ by ,b ,e2 !n half ( x1 or ,b ,a ,c1 mu/ be measur$ by half ,b ,e4 ,9 l mann]1 x may be prov$ t ! angle ,c ,a ,d is measur$ by half ,e ,d4 ,!n1 because ,b ,a ,c is measur$ by half ,b ,e1 & ,c ,a ,d by half ,e ,d1 ! :ole angle ,b ,a ,d mu/ be measur$ by half ! :ole arc ,b ,d4 ,,second ,,case4 ,suppose ! angle 7 :olly on "o side ( ! c5tre1 z ,f ,a ,b4 ,draw ! diamet] ,a ,e & ! radius ,b ,c z be=e4 ,prove t ! angle ,b ,a ,e is measur$ by half ! arc ,b ,e4 ,draw ano!r radius f ,c to ,f1 & prove t ,f ,a ,e is measur$ by half ! arc ,f ,e4 ,!n1 because ! angle ,f ,a ,e is measur$ by half ! arc ,f ,e1 & ! angle ,b ,a ,e is measur$ by half ! arc ,b ,e1 ,! diff]5ce ( ! angles1 or ,f ,a ,b1 mu/ be measur$ by half ! diff]5ce ( ! arcs1 or half ( ,f ,b4 ,illu/ration ,,proposition ,,xxi4 ,,!orem4 ,,demon/ration4 ,we wi% to prove t ,p>allel *ords int]cept equal arcs ( ! circumf]5ce4 ,let ! *ords ,a ,b1 ,c ,d1 be p>allel2 !n w ! int]cept$ arcs ,a ,c & ,b ,d be equal4 ,= draw ! /raight l9e ,b ,c4 ,because ! l9es ,a ,b & ,c ,d are p>allel1 ! int]ior alt]nate angles ,a ,b ,c1 ,b ,c ,d1 are equal4 ,b ! angle ,a ,b ,c is measur$ by half ! arc ,a ,c2 ,& ! angle ,b ,c ,d is measur$ by half ! arc ,b ,d3 ,!n1 because ! angles are equal1 ! half arcs :i* measure !m mu/ be equal1 & ! :ole arcs !mselves mu/ be equal4 ,illu/ration ,,proposition ,,xxii4 ,,!orem4 ,,demon/ration4 ,we wi% to prove t ,! angle =m$ by a tang5t & a *ord meet+ at ! po9t ( contact is measur$ by half ! int]cept$ arc4 ,let ! tang5t ,c ,a ,b & ! *ord ,a ,d meet at ! po9t ( contact ,a2 !n w ! angle ,b ,a ,d be measur$ by half ! int]cept$ arc ,a ,d4 ,= draw ! diamet] ,a ,e ,f4 ,because ,a ,b is a tang5t1 & ,a ,e a radius at ! po9t ( contact1 ! angle ,b ,a ,f is a "r angle1 & is measur$ by ! semicircle ,a ,d ,f4 ,because ! angle ,f ,a ,d is at ! circumf]5ce1 x is measur$ by half ! arc ,d ,f4 ,!n ! diff]5ce betwe5 ! angles ,b ,a ,f & ,d ,a ,f1 or ,b ,a ,d1 mu/ be measur$ by half ! diff]5ce ( ! arcs ,a ,d ,f & ,d ,f1 or ,a ,d2 ,t is1 ! angle ,b ,a ,d is measur$ by half ! arc ,a ,d4 ,illu/ration ,,proposition ,,xxiii4 ,,!orem4 ,,demon/ration4 ,we wi% to prove t ,a tang5t & *ord p>allel to x int]cept equal arcs ( ! circumf]5ce4 ,let ,a ,b be tang5t to ! circumf]5ce at ! po9t ,d1 & let ,c ,f be a *ord p>allel to ! tang5t2 !n w ! int]cept$ arcs ,c ,d & ,d ,f be equal4 ,= f ! po9t ( contact ,d1 draw ! /raight l9e ,d ,c4 ,because ! tang5t & *ord are p>allel1 ! int]ior alt]nate angles ,a ,d ,c & ,d ,c ,f are equal4 ,b ! angle ,a ,d ,c1 be+ =m$ by ! tang5t ,d ,a & ! *ord ,d ,c1 is measur$ by half ! int]cept$ arc ,d ,c2 ,& ! angle ,d ,c ,f1 be+ at ! circumf]5ce1 is measur$ by half ! arc on :i* x /&s1 ,d ,f3 ,!n1 because ! angles are equal1 ! half arcs :i* measure !m are equal1 & ! arcs !mselves are equal4 ,illu/ration ,,proposition ,,xxiv4 ,,!orem4 ,,demon/ration4 ,we wi% to prove t ,! angle =m$ by ! int]section ( two *ords 9 a circle is measur$ by half ! sum ( ! int]cept$ arcs4 ,let ! *ords ,a ,b & ,c ,d int]sect ea* o!r 9 ! po9t ,e2 !n w ! angle ,b ,e ,d or ,a ,e ,c be measur$ by half ! sum ( ! arcs ,a ,c1 ,b ,d4 ,= f ! po9t ,c draw ,c ,f p>allel to ,a ,b4 ,because ! *ords ,a ,b & ,c ,f are p>allel1 ! arcs ,a ,c1 ,b ,f1 are equal4 ,add ea* ( ^! equals to ,b ,d1 & we h ,b ,d plus ,a ,c equal to ,b ,d plus ,b ,f2 t is1 ! sum ( ! arcs ,b ,d1 ,a ,c1 is equal to ! arc ,f ,d4 ,because ! *ords ,a ,b1 ,c ,f1 are p>allel1 ! opposite ext]ior & int]ior angles ,d ,e ,b1 ,d ,c ,f1 are equal4 ,b ,d ,c ,f is an angle at ! circumf]5ce1 & is !re=e measur$ by half ! arc ,f ,d4 ,!n ! equal angle ,d ,e ,b mu/ be measur$ by half ( ! arc ,f ,d1 or its equal ,b ,d1 plus ,a ,c4 ,illu/ration ,,proposition ,,xxv4 ,,!orem4 ,,demon/ration4 ,we wi% to prove t ,! angle =m$ by two secants meet+ )\t a circle is measur$ by half ! diff]5ce ( ! int]cept$ arcs4 ,let ! secants ,a ,b1 ,a ,c1 int]sect ! circumf]5ce 9 ! po9ts ,d & ,e2 !n w ! angle ,b ,a ,c be measur$ by half ! diff]5ce betwe5 ! arcs ,b ,c & ,d ,e4 ,= f ! po9t ,d draw ! *ord ,d ,f p>allel to ,e ,c4 ,because ,a ,c & ,d ,f are p>allel1 ! opposite ext]ior & int]ior angles ,b ,d ,f & ,b ,a ,c are equal4 ,because ! *ords ,d ,f1 ,e ,c1 are p>allel1 ! arcs ,d ,e & ,f ,c are equal4 ,if f ! arc ,b ,c we take ! arc ,d ,e1 or its equal ,f ,c1 we %all h left ! arc ,b ,f2 ,b ! angle ,b ,d ,f1 be+ at ! circumf]5ce1 is measur$ by half ! arc ,b ,f3 ,!n ! equal ( ,b ,d ,f1 or ,b ,a ,c1 mu/ be measur$ by half ! arc ,b ,f1 or half ! diff]5ce betwe5 ! int]cept$ arcs ,b ,c & ,d ,e4 ,,app5dix4 ,,note ,a4--,to ^? tea*]s :o ?9k t ! l9e %\ld be d]iv$ f a surface1 & ! surface f a solid1 ! au?or w\ld say1 t1 accord+ to 8 exp]i5ce1 *ildr5 appreh5d ! ideas convey$ by ! t]ms l9e & surface z readily z !y d any ideas :atev]2 & t1 !re=e1 !re seems to be no necessity = extraord9>y c>e 9 ?is case to avoid giv+ wrong impressions4 ,/ill1 if x be consid]$ desirable 9 ?is mann] to d]ive l9es & surfaces1 x w be app>5t t all t c be done 9 ! matt] is to give su* in/ruction only by way ( a prelim9>y lesson4 99 99 99 99 99 ,,note ,b4--,crook$ & curv$ l9es are "h treat$ ( be=e /raight l9es1 because ! fir/ two are def9$ by means ( an affirmative prop]ty1--!y d *ange direction2 :ile ! la/ is def9$ by means ( ! abs5ce ( "o1--!y d n *ange direction4 ,x is easi] = a *ild to compreh5d :at is ?an :at is n4 99 99 99 99 99 ,,note ,c4--,if ! pupils are old en\gh1 !y may be %[n t v]tical l9es _c be p>allel1 b only seem s on acc\nt ( _! %ortness & ne>ness to ea* o!r4 99 99 99 99 99 ,,note ,d4--,?is def9ition may be consid]$ objectionable because rhomboid means l a rhomb4 ,t ! m g5]al figure1 ! rhomboid1 has be5 nam$ f ! m re/rict$ "o1 ! rhomb1 is un=tunate1 because x int]f]es ) ! symmetry ( ! nom5clature4 ,! rhomb possesses all ! prop]ties ( ! rhomboid1 & %\ld1 !re=e1 :5 ^! are consid]$1 be call$ by ! same "n2 its additional prop]ty entitles x to a "n :i* %\ld compreh5d ! o!r names4 ,if ! rectangle _h be5 call$ a squ>oid1 ! difficulty w\ld h be5 repeat$4 99 99 99 99 99 ,,note ,e4--,if tea*]s consid] x desirable1 !y may require ! class to prove1 by way ( coroll>y1 su* propositions z ass]t ! p>allelism ( ! l9es :5 ! int]ior alt]nate angles are equal1 :5 ! opposite ext]ior & int]ior angles are equal1 c41 9 cont9uation ( :at has already be5 done4 ---------------------------------------- -------------------------------- ,,t[ns5d',s ,,analysis ,,( ,,! 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