350 RELATION BETWEEN THE AREA OF A PLANE 
to differ by any finite angle from two right angles. In like manner, 
if it be observed that CF is gyeater than the sum of the lines, AC or 
CK, KN, rh, &e., it will appear that s, does not ultimately differ by 
any finite angle from two right angles. Therefore. ultimately the 
quantity, S,—s,, is less than any assignable angle. But it was 
proved that 
S,—s, =S—s. 
Therefore 8 and s do not differ by any finite angle; that is, they are 
equal to one another. 
Cor. 1.—If two triangles ACB and FCD, having the angle ACB 
equal to the angle FCD, be unequal; and ACB be the greater ; then 
S, the sum of the angles of the triangle ACB, is not greater than s, 
the sum of the angles of the triangle FCD. For, the same construc- 
tion as that described in the Proposition may be made, until a pomt 
is reached at which one of the triangles obtained, as Thr, has the. 
sides, Th, kr, either less than Lh and AF respectively, or greater than, 
Lh and hF respectively. The former of these cases cannot occur :, 
because then the triangle Thr would be less than the triangle FAL, 
and consequently the triangle ACB less than the triangle FCD :- 
which is impossible. Hence the latter case must oceur, viz.: that a 
triangle Thr must be found, having T/ greater than AL, and rh. 
greater than 2F ; and therefore, since the triangle F/L can be wholly. 
inserted in the triangle Thr, the sum of the angles of the triangle 
Thr is not greater (Cor. 2, Prop. 1.) than the sum of the angles of 
the triangle FAL. Hence S is not greater than s. 
Cor. 2.—If two equal triangles (Hig. 9) 
ACD and BCD have the common base CD, 
and if S be the sum of the angles of the 
former, and s the sum of the angles of the 
latter, S is equal to s. For tne difference 
between S and s is the same as the differ- 
ence between the sum of the angles of the | 
triangle ACE and the sum of the angles of i 
the triangle BDE. But, by the Proposition, these latter. quantities 
are equal to one another. Therefore S=s. 
Cor. 3.—Let the two triangles (see fig. to Cor. 2) ACD and BCD, 
on the common base DC, be unequal. Then, if S be the sum of the 
angles of the triangle ACD, and s the sum of the angles of the. tri- 
