687 / 1042
o
~
l ..
~ tri~nr.l<
hRVirr, . JI ils ,hrce
(i~l~,qu,1
:000<
~no·
t}.lf,
is
-::.11t6i AI1
(fUrl.!d'''¡ 11;111.',.:1,•
.
IS.\,
li~.
Z5.
<.',
t\
uldn,;h: hotV,ug 1"'0 uf ¡'s
li..
:~s
ltltl.d
lO 0.1(' a·
nllllh.' f.
:.ml
Ihé lhird
¡iJ,'
nuC ('(¡U,
I
{l'
cidlt r
uf
(i¡un,
1)
~J lIl,1
dO
'f/ú1o
Irinll,~/(1
ilS
H,
fi:~ ~6.
4;' A (t"ngle h,vinC none of il! I:,!cs '·qu.,1
10
on~
~nOl"".
is e.,lIeo a
fi:o/"" Iriongh.
as C. fig
27.
4'1'
Tri.n"I« . \\'ilh rdpeél
10
their .ngles. are dividtd
in,~
Ih" . rl,O"renl kinds ,
,·iz.
4í. :\.
,nan~le hav'll~
(lnc
01'
its .ncles righl. is called
a".
hl 'O"g/fJ
Irtdllfl..,
,s
A,
fig.
~g .
46.
t\
Imncl~ I""i,,~
one (,f
lIS
.nglrs oblufe, or
gr.,lrr tI,dn • ngh l
ar.~!e.
is called an
./o/uft·.ng/ed Irt·
QI:;;'(,
~s
11.
lig.
'29.
47 Lalfly,
a
tn, ngle h,ving all ils arogles aeulc. is
c<ll:c:d
An
(1(lIt('ll1~~hd
!r¡,";glr,
as
el
{¡:.!.
~o.
43.
In all
ri~ht.ar.g;lll
trl<"Aks,
ti,.,
(¡clc, comr'"
h'IlJ¡n~
Ihe
I
ighl
.n~1c
are e.Jled Ihe
le):,.,
.od Ihc fidc
orpolil< lO the tighl , ngle i, "lIfd Ihe.
'),/",¡ I,,·n ul ' .
1'liu5
,n Ihe rig'll.,ncled mangle JI
Be,
fig.
3(.
(Ihe right
'''Sic bciog ¡.t H) Ihe til.'Olid,s AB .oJ
BC,
wl,ieh COI11'
pl ehcnd ,he
ri~h t
angle A!:lC.
"r~
the lec. uf Ihe Iriall'
gle ;
.lId
,loe liJe
tiC'.
\\'hi,h is ol'pofile
lO
Ihe ritht
~o·
gle.l,H • is Ihc h
y~lIIhe,,"le
of the richl.anglcd tri.oSlc
.~BC.
49. Roth obtuCe ao l acule
an~l~d
tri.ngb are io
~c·
oeral callfd
.bli1," ·al/g/td Iri'!IIg/o;
in all whieh
~ny
fiJe is called ¡he
boj:,
.nd Ihe olher 1100
IhejidtJ.
50. T he perpendicolar
hci~ht
of any tn.ngl: is
A
linc
d"wn from Ihe vertex
10
Ihe b. Ce perpendlcll larly ; Ihus
if Ihe trianglc AfiC (fig.
3
~ .)
be propo(cd, ,nd IlC be made
ils
~&.
Ihtn A will I'e Ihc venex.
vi:.
Ihe aogle oppofilc
(O
Ih~
b,II".:; .mJ if (rom
A
you draw the hne AD per.
per.die" l«
(O
HC, Ihen Ihe line
!I D
is Ihe heiglil of Ihe
trl,"gle tlHe Ilandlng on !:lC as its bale.
Hen,c
.H
!lianglcs flandlOg belwten the (. me parallels
hlVC the
(
.r.lehe,~hl,
lince all Ihe perpendiculm are
c·
qu.1 by Ihe nalore of parallds.
5l.
A
lig~re
bounded by four fides is called a
fuadri·
11/<,,/
or
flladral/!" I"r/ixur"
as
tI!:lDC,
fig
33.
52 .
~Hlrildlerdl fi~ures
who(c oppolil< lides are pa·
r.llt1 .
are
ealled
par.llt/'gral/II.
T hus in Ihe quadnla.
Im l filiore :\l:lDC. il the lide:\.C be paralid
ti>
the fide
!ll) willth i, oppolile
10
il, and Aa be paralld
10
CD,
I:,en Ihe fig'lre
ti
HOC is calleo a pmllelogram.
53 .
ti
pmlldogram h.lYing
~II
ilsfiJeseqll,,1 ,ntl ancl,s
right, i, ealbl a
('luan,
's
ti ,
fiC·
34.
54. Th<t "hidlll3lh only Ihe oppofile fides equal aod
its dnfl,s righl. is c.llled a
rol/al/g/' ,
as
H,
lig
35.
5i· Th"t ..hieh halh equ,1 fid"
b~t
ohllqoe
30~lrs,
is
ealled
a
rh'lI/bu/,
dS
C.
fi~
;6.
and IS joll an iochned
(
qu.re.
56. Thn ",hieh h"h only Ihe oppofi,e
fi,lc~ equ~1
, 0J
Ihe
~ngl<l
obJ¡que. is ,,1I,d a
rh.m¡',,,/,,.
•
s 1).lig. 37.
and m.y be coneei ' cd 's 'o inehne"
reé"'n~ll·.
57 \Vhen none of th, fides >le p:lldlU 10 anolher.
Ihen Ihe
qu~d"later,1
figure is
eaJlcd. I,"p<:iuOll.
58. Every other righ l IIOeJ
fi~ure,
tI!JI h,s more fidcs
than fOIIl, is in
~encral
e"lkd a
p'/,.~.n.
AnJ
fi~III"<
.Ir.
c.alkd
by
pJllicul.trnameS
ilCl:wJ'II}!
ICI lhe:
nun~!llr
oC
Ibr fiOlS,
,.¡, .
one
01'
li,e liJ,s i. c.I"J.
/,f"I,'.~ .I/,
of
T
n
Y.
(,;:7
f~):
ílll),::;:M,
(lf
Ic\'cnrt
ht':'.'(~~
':,
ard fa ('n.
\f!t n
lh· 1;, It,
!'orrt:lOl:
dle:
pIIIYJ.flO, re
(:C¡U:..lloúl.\.
a.<:lh'.I) lhe
fig'lrc is <..dItU.i a
rchd
Ir
hf'lrc or
P(,!~·~(¡n .
'9 ,
hc.'~'
n
¡ti
I
!.;/t'
I\ I ¡~:
(ht!.
~~.)
0,'.'( r/ilJ
'::J,OJ
];(:
/J'I1.'f.
I'rr:dur:d " f",t'ol'd,
O,
/ht
u
/en.alQJ.:,/~
ACD
IJ
11'"
// r;
h,Jl,¡(
JI: /!'
':dl
(I/t'ji!(
0110
lok"1J
';Q.
gd/ofr.
,·iL.
1,
AI' e
0/,''/
fJ
~c.
lovtder
10
prove I"'S.
Ihr('ugh
C
d ,~w
CE
p"r~i ld
10
Al:: then finee CE '"
poraJlel
to
Al!.
aod the
I;t.l ~
¡\C .nd
HD
crofillh Ihcm,
Ihe
an~lc
ECD is equal
rO
AIlC (by an .
36.)
ar.d Ihe
aogle "CE
tg~, 1
:0
Clill (b)' Ort o
35.) ;
It,m:ore the
angles ECD aod ECA arc equ.! tO toe . rgl(s A!:lC and
C:II:l; bUI Ihe onglós ECO aod ECA
<re
locelher
e–
qual 10 the ,nglc ACJ) : Ihnefore Ihe angle ACD
i~
equal lO bOlh Ihe angles AIlC and CA Il laken IOgelher.
C.r.
Hence il m, y he pro\'ed, Ihal if two lioes
AB
.nel
Cl) (fig.
39')
hc crotred hy a Ihird lir.e EF, and Ihe al–
leroale dngles AH' ,0,1 EFl) be equal. Ihe (¡nes AIl and
CI) will be p:lraJld; lur il' Ihey are nOI parallel. Ihey
I'Hlfl
mw one anOl hor "U ooe fide of Ihe line
Er
(Cup–
POI( al G) aod fo form Ihe tri.ngle
E.fG. 00: of whofe
~dcs
CE
beiog proJuced tO
t\.
the eXlerior ancle AEF
mufl
(b)' this :mirle) be equal lO Ihc fum of Ihe t\Vo
a~gJcs
EFG and ¡·.GF; bUI, by (nppofilion, il isequal lO
Iht angle EFG alolle; Ihmfore the
.n~lc
.'I.E F mui! be
eo,ual
10
Ihe fumof Ihe I\YO angles EFG and EGF, anel
at Ihe fame lime e'I".1lO Ihe .ngle EFG 3100e. whieh
is dbfuro; fo Ihe "nes
1\13
aoo CD
ear.oo!mw,
aml
Ihcreforc mull bc p.ralle!.
60
In
ar.y
Il'ian~/,
ABe,
all
ti·,
1.lru
angla
la.!",
I<gelh" arr
,qua/ID
loO?
ri,eh ang/-I.
To prove Ihis,
yuo
m~n
produce lJ C, ooe of ilS legs, lO aoy dia. nce,
(uppofc tO O ; Ih<o by the lan propofilion. !be eXlern.l
at>gle, ACD. is t9ual
10
Ihe fum of Ihe IWO iOlernal
0pl'0file ones C:I
i:
and AUC;
10
bOlh ,dd Ihe aogle
AC13, Ihen Ihe Cum of Ih: anglós ACD and ACH will
be c'lu1110 Ihe (um of Ihe angles
C
AB aod CIJA and
tlCB . HUI the fu mof Ihe angles ACO anJ ACB, is
etlu.11lO I\VO
tÍ~ht
ones (by art o
3~
). Ihercf."e Ihe (um
of thc time , ngles CAB .nd CH.\ apd ACB. is equal
lOIWO righ l
ar.gb; thalls, Ihe fum of Ihe t!me a"gles
of a"y " iangle tlC:l is .qu. 1 lO IWO righl .ngb.
e...
t.
I
j.·nee
10
any triangle
~i,en.
if one of ilS
aogiLs b.
~no\Yn,
the fum of Ihe olher l\\'O i$ a:Co
known: for lince
(by
Ih:
I~II)
Ihe (umof all Ihe three is
equll lO IWO righl .ng!rs, or a rClOieircle, il i; plain, tha!
taking any one of them imlll a femitircle or 180
de~r<cs,
Ihe
rcm.,¡nJ~r
\\'¡I
l-e
thc lum oi Ihe o:hcr 1\\'0. T hus
(io Ihe fOI!llt< trlau¿le :I IlC) iC the angle AIlC bc 40'
negrees.
~y
14kinr, 'lo Irom I
S.:>
'c have 140 degrees:
which is Ihe fom of Ihe IWOongks B.\C• .
"CI3:
Ihe
cOO\'clf, of Ihis is alfo
pl.in. t,i:. the rom of .ny l\\'O
:tn~lc.'s
('(
a tri,Hln1c
Ut·i,'}!
f., \ ..
II,
tlu;> othér
:.Ingle!
is Ah\)
~tlo\Yn
by
lakln~
Ih.. C,,,n r,om I
$0
,je~rees
.
2.
In aoy
"r.hl.an~I ,J trian~I
•• Ihe 1"0 acule angle$
muJI jull mak, "p a
,,;~hl
one brlw!'en th,01;
eonf"'I ,,~nl'
1)', any oOe (lf Ih. olohqut angles b.ing gi",·n.
\1,
n,,¡y
fin,1 Ih. oliler h
Iillllralilll~
the gil'cn uoe f, om
~J
de·
&". ,
\\'I",h
i~
Ih,' fUlc. of bOlh.
lit.
¡I'in ony
1 '\O"ri~n¡:lrs
•.
~ !l C :;~
..
lC'.~
DEF (Ii&.
41.) 1110
h~s
¡.(
Ih<
,111•• "': .
:\ü"J
.\C,~,
,",u,1 In
"\ v