ye ch 
° 
‘sible, (Ax. 9.) 
_ away from each the common 
“sides which 
GEOMETRY. 
For their sum is equal to the two angles BAD and 
DAF, which together make two tight angles. 
Cor. 3. (Fig. 26.) All the ‘which any num- 
ber of lines AB, AC, AD, AE, &c. make about a point, 
are equal to four right angles.” For through A draw 
a straight line PQ, then all the angles which the 
lines make on each side*of PQ, are equal to two right 
angles: therefore, all the angles on both sides, which 
make up the angles about A, are equal to four right 
“yA. ahi fe , nif"; 9 , : 
angles, 
tse ff oa “Prop. IL. Turon. . ' 
If two adjacent angles ACD, DCB are together equal 
to two right angles, the two exterior sides AC, CB form 
one continued straight line. ' | 
For, if CB be not the continuation of CA, ‘suppose 
CE to be its continuation ; then the sum of the angles 
ACD, DCE is equal to two right angles, (2.) But by 
hypothesis, the sum of the angles ACD, DCB is equal 
to two right angles ; therefore ACD4 DCE=ACD+4 
DCB (Ax. 1.) And taking from each the angle ACD, 
there remains the angle’ DCE equal to the angle DCB 
(Ax. 3.), a part equal to the whole, which ts impos- 
ates Prop. IV. Turon, wr ye 
Tf two straight lines AB, DE cut each other, the ver- 
tical or opposite angles shall be equal. 
’ For because DE is a straight line, the sum of the 
two angles ACD, ACE is equal to two right angles, 
(2.)3 and because AB is a straight line, the sum of 
e angles ACE, ECB is equal to two right angles, 
(2.); therefore the sum of the angles ACD, ACE is 
equal to the sum of the angles ACE, ECB ; and taking 
; angle ACE, there remains 
ie ie ACD equal to the vertical or opposite angle 
Prop. V. Tueor, 
Two straight lines which have two common points 
coincide entirely throughout their whole extent, and 
form but one and the same straight line. ; 
Let A and B be the common points; in the first place, 
the two lines can make but one, from A to B (Ax. 10.) 
If it were possible that they could separate at C, let us 
U that the one takes the direction CD, and the 
the direction CE. At the point C, suppose CF 
to be drawn perpendicular to AC ; then, because ACD 
is by esis a straight line, the angle FCD is a 
right angle, (Def. 9.) ; in like manner, because ACE 
is supposed a straight line, the angle FCE is a right 
angle; therefore, the angles FCD, FCE are equal, 
(1.); but this is impossible, (Ax. 9.) ; therefore, 
the straight lines which have two common points A, B 
cannot separate, but must form one continued line. 
Prop, VI. Turer. 
Two triangles are equal, when an angle, and the two 
contain it in the one are equal to an angl 
VOL, X. PART I, 4 ower 
201 
and the two sides ‘which contain it in the other; each 
to each, 
Let the angle A be equal to py D, the side 
AB equal to the side DE, arid the side AC equal to the 
side DF ; the triangles ABC, DEF shall be equal. 
Suppose the triangle ABC to be placed upon the 
triangle DEF, so that AB may be on DE ; then, be- 
cause the angles A and D are equal, AC will fall on DF, 
and because AB=DE, and AC=DF, the points B, C 
will fall on the points E, F; therefore the base BC will 
coincide:with the base EF (5.); and the triangles will 
coincide entirely ; therefore they are equal, (Ax. 8, 
Cor. Hence it follows, that the bases or third sides 
BC, EF are equal, and that the remaining angles B, C 
of the one, are equal to the remaining angles E, F of 
the other, each to each, viz, those to\which the equal 
sides are opposite. 
Prop VIL. 
Two triangles ate equal, when a side and two adja- 
cent.angles of the one are equal to a side and two adja- 
cent angles of the other, each to each. 
Tueor, 
Let the side BC be equal to the side EF, the angle 
Principles. 
—_—y/ 
Fig. 30. 
Fig. 30. 
B equal to the angle E, and the augie C equal to the. 
angle F ; the triangles shall be equa 
suppose the triangle ABC to be placed upon 
DEF, so that their aud bases BC, EF may coincide; 
then because the, angles B, E are equal, the lme BA 
will fal] on ED; a hcbainse the angles C, F are equal, 
the line CA will fall on FD ; therefore the three sides 
of the one triangle will coincide with the three sides of 
the other, and the triangles will be equal. 
Cor. Hence it appears that the remaining angles 
A, D of the triangles are equal, and that the remain- 
ing sides AB, AC of the one are equal to the remain- 
ing sides DE, DF of the other each to each, viz. those 
to which the equal angles are opposite. 
Prop. VIII. 
Any two sides of a triangle are together greater than 
the third, 
THEOR. 
For, in the triangle ABC, the straight line BC is the 
shortest line that can be drawn from B to C, therefore 
BC is less than BA+- AC. 
Prop. IX. Tueor. 
If from any point O within a triangle ABC, straight 
lines OB, OC ‘are drawn to-the extremities of the base 
BC, their sum is Jess than the sum of the two sides 
AB, AC.. . 
' Produce BO until it meet AC in D; the line OC is 
less than OD + DC, (8.), and adding to these unequals 
BO, we have BO4O0C— BO+OD + DC, (Ax. 4.) ; 
that is BO4+OC — BD + DC. 
In like manner, BD.— BA + AD; and adding DC, 
BD + DC = BA + AD + DC, that is BD + DC _=— 
BA-+AC ; but we have found BO4+ OC— BD+ DC; 
much more then is BO 4+ OC BA + AC. 
Prop. X. Tueor. 
If two sides AB, AC of a triangle ABC are equal to 
Ze 
Fig. 30. 
Fig, 31. 
Figs. 32, 
33, 34. 
