TRIANGLE AND THE SUM OF THE ANGLES. 347 
variation, until DC is less than VB. Then ultimately the triangle ADC 
may be wholly inserted (as in Fig. 7) within the triangle EWA. For, 
since the sum of the angles of the triangle ADC falls short (by hy- 
pothesis) of two right angles by more than the angle BAH, the angle 
DAC must be ultimately less than the angle BAW ; and therefore DA 
falls between BA and WA. Again, the point D cannot lie beyond EW ; 
else DC would be greater than the perpendicular from C upon EW, 
and consequently (since AC is less than AB) greater than BV: which 
is contrary to hypothesis. Hence (Cor. 2, Prop. I.) the sum of the 
angles of the triangle ADC is not less than the sum of the angles of 
the triangle EWA But the sum of the angles of the triangle ADC 
is (by hypothesis) less than the angle EAW: which is impossible. 
Consequently, as DC diminishes indefinitely, neither of the other 
sides, AD, AC, becoming at any stage greater than AB, the sum of 
the angles of the triangle ADC cannot ultimately differ from two 
right angles by any finite angle. 
Proposition IV. 
If ABC and FCD (Fig. 8) be two triangles of equal areas, and 
having the angle ACB equal to the angle FCD ; and if S be the 
sum of the angles of the triangle ABC, and s the sum of the angles 
of the triangle FCD; S and s are equal to one another. 
For, if the sides FC and CD be equal to AC and BC, each to each, 
the triangles ABC and FCD are equal in every respect. It is there- 
fore only necessary to consider the case in which FC is greater than 
AC: im which case (in order that the triangle ABC may not be a part 
of the triangle FCD) CD must be less than BC. Place the triangles 
so that AC and CF may be in the same straight line; in which case, 
since the angle ACB is equal to the angle FCD, BC and CD are in 
