GEOMETRY. 
arc apply (the confequent) AC from A to c; draw Ac, 
and -ttpply c A from c to D. Apply the following lines 
in the order direfted, vist. A D from 
A to (fy and from to E ; A E from 
A to r, and from t to F ; A F from 
A lo/y and from/to G ; A G from 
A to^, and from^toH, 8 cc. Then 
will tlie proportional lines be A B, 
ACj AOjAE, Af'jAGj&c. And 
AB:AC::(Ac = )A'C:(At/=:)AD. 
AB:AC::(Afl'r=)AD;(Ar=r)AE. 
AB:AC::(Ae=::)AE: (A/=)AF, 
EC D 
&c.—This depends on articles 104, 95, 165, 
THEOREMS, 
Of Right Lines and Planes. 
91. JV/ien cm n'g/it line C D /lands upon another right line 
A B, they make two angles BCD, A C D, which together are 
equal to two right angles.- — Demcn/lra- 
tion, P'or, deicribing a lemicircle 
A D B, on C (45) ; then tire arc D B 
meafures the angle B C D (15); and 
the arc D A meafures tlie angle 
A C D (15) ; but tile arcs D B and 
■“AD together meafiire two right 
angles (13, 16) ; therefore BCD and A C D together, 
a-re equal to two right angles (50). 
92. Corollary. Hence if any number of right lines Cd 
Bands on one point C, on the fameTide of anotlier right 
line A B ; the fum of all the angles are equal to two 
right angles ; or are meafured by 180°. 
93. If two right lines, AC, EB, intcrfeEl each other in D, the 
oppofite angles are equal, viz. ^ C D B 
= 2iADE, and ^CDE=zADB. 
For the angles A D E and A D B to¬ 
gether make two right angles (91); 
and the angles C D B and A D B to¬ 
gether make two right angles (91); 
therefore the fum of ADE and ADB 
fum of C D B and ADB (46) ; 
confequently the ^C DB is.equal to the A D E (48). 
94. Coral. Hence if any number of right lines crofs 
each other in one point, the fum of all the angles which 
they make about that point, is equal to four riglit angles; 
or is meafured by 3(10°. 
9'3. If a right line FE cut two parallel right lines, AB, 
C D j th«n is the outward angle a equal to the' inward and oppo- 
y , file angle d ; and the alternate angles, c, d, 
are equal: and the contrary .—Becaule 
C (a. j) C D and A B are parallel by fuppo- 
lition ; then F E has the fame incli¬ 
nation to C D and A B ; and this in¬ 
clination is exprefl'ed by the /_a or 
/_d-, therefore the outward is 
equal to the inward and oppolite 
^d. Now the is equal to thc/;e. 
And fmce the ^ a is equal to the ^ d\ therefore the al¬ 
ternate angles c and d are equal. 
96. hi any right-lined triangle 
ABC, the fum of the three angles 
a, b, c, is equal to two right angles: 
And if one fide B C be continued, the 
outward angle f is equal to the fum 
of the two inward and oppcfite angles 
a, b. —Through A,draw ariglu line 
parallel to BC, making with AB 
the fe, and with A C the /_d. 
Liow being-alternate. 
And two right angles meafure the /.e ^ fd. 
Therefore Z.bAr.fo.-\-;fcz=iZ.o-\-j<,a.\-y/d. 
Confequen-.ly Ab ^ /.a /_c— two right angles. 
Moreover Z<^-lrZ/ = two right angles,.' 
Therefore — Af’ 
VoL.VIII, No.514. 
97. Hence, if one Angle is right or obtufe, each of the 
other is acute. 
98. If two angles of one triangle are equal to two 
angles of another triangle, the remaining angles are 
equal. And if one angle in one triangle is equal to one 
angle in another triangle, then is the fiim of the remair. 
ing angles in one, equal to the fum of the remaining 
angles in the other. 
99. If two ft dcs A B, AC, and the included angle A, 
one triangle A B C, are refpeclivcly equal to twoftdes D E, D F. 
and the included angle D, of another 
D E F, each to each ; then arc thofe 
triangles congruous. —Apply the 
point D to tile point A, and the 
line D E to A B. No-.v as D E ee: 
A B (by fuppofilion) ; therefore 
tlie point E falls on B. But 
ZD=2ZA (by flip.); therefore 
D F will fall on AC. And fince 
DF=:.AC (by flip.); therefore 
the point F Bills on C. Confequently F E w ill fall on 
CB; therefore the triangles A C B, DFE, are con¬ 
gruous, fmce every part agrees. 
too. If two triangles ABC, D E F, have two angles A, B, 
and the included fide AB, in one, refpeBively equal to two angles 
D, E, and the included fde T)¥,, in q -p 
the other, each to each ; then are thofe 
triangles congruous. — Apply the 
point D to A, and the line DE 
toAB. NowasDE=;AB (by 
flip.); therefore the point E falls 
onB. And asZD = ZA (by 
flip.); therefore the line D F falls 
on A C. Now if the line A C is 
lefs or greater than the line D F; 
then the line F E not falling on C B, makes the Z ^ 
or greater than Z E- But Z B — Z E (t>y ft'P-)j there- 
A C is neither lefs nor greater than D F. Or the line 
A C D F; confequently F E = C B ; therefore the tri¬ 
angles are congruous. 
-X 
B :d. 
101. Two triangles, ABC, D E F, are congruous, 
the three fides in the one are equal to the three ftdes in the 
each to each .—Apply the point D 
to A, and tlie line DE to AB. 
Now as D E 5= A B (by fup.) ; 
therefore the point E falls on B. 
On A, with the radius AC, de- 
fcribeanarc. Then as DF AC, 
the point F will fall in that arc. 
Alfo on B, with tiie radius B C, 
deferibe another arc, cutting tiie 
former in C. And fmce E F B C, tlie point F wil 
in tliis arc alfo. But if the point F can tall in both 
arcs, it can be only wiiere they InterfeCl, as in C. 
fequcntly the triangles arc congruous, or identical. 
102. Two triangles ABC, D E F, are congruous, when 
two angles A, B, and o fide A C oppof-te to one of them, in one 
triangk, are refpeSlively equal to two 
angles D, E, and a fde DF o/ipofte 
to a like angle in the other triangle, 
each to each .—Apply the point D 
to A, and the line D F to AC. 
Now as DF—AC (by fup.); 
therefore the point F tails on C. 
And as Z D = Z ^ (*^y fuP-) 1 
the line DE will fall on the line 
AB. And if the point -ti does ■ 
not fall on B, it mu it fall on fomeotlier point G. Draw 
C G. Then the angle AG C is equal to the angle D E F. 
And the angle A B C 2= (D E F AGC, winch is 
not pofiible ; therefore tlie point h, can fall no where 
but on the point B. Coniequently the triangles are 
congruous. 
• 5 R -f” 
