TRIANGLE AND THE SUM OF THE ANGLES. 351 
angle BCD, and if the former triangle be greater than the latter, S 
cannot be greater than s. For the difference betweea S and s is equal 
to the difference the sum of the angles of the triangle ACE and the 
sum of the angles of the triangle BED. But the former of these 
quantities (since the triangle ACE is greater than the triangle BED) 
is not greater (Cor. 1) than the latter. Therefore S is not greater 
than s. y 
Cor. 4.—In the case supposed in the previous Corollary, should 
the assumption be made that the angles of a triangle are not (see Cor. 
Prop. IJ.) equal to two right angles, S must be less than s. For, by 
the reasoning in the Proposition and in the foregomg Corollaries, it 
appears that the difference between S and s is equal to the difference 
between the sum of the angles of a triangle ACB (Fig. 10) and the 
sum of the angles of a triangle ADE in- 
scribed withim the former in the manner EIG.IO 
shown in the figure. Suppose, if possi- 
ble, that S=s. Then the angles of the 
triangle ADE are together equal to those 
of the triangle ACB. Therefore (Cor. 1. 
Prop. I.) they are equal to those of the 
triangle ACE. ‘Therefore angle ADE is a = B 
equal to the suin of the angles DCE and 
DEC. ‘Therefore the angles of the triangle DEC are together equal 
to two right angles: which is at variance with the hypothesis on 
which we are at present proceeding. Hence § is not equal to s. 
But (Cor. 3) 8 is not greater than s. Therefore S is less than s. 
Cor. §.—If the triangle ABG (Fig. 11) be divided by the straight 
line AC into two parts, of which ACG is the greater, two lines AD 
A 
a 
