A. C. Twining—EKuclid's Doctrine of Parallels. 335 
Prop. XXVIII 4. THEOREM. 
No triangle can have the sum of its angles greater than two rught 
gles, 
Let ABC be atriangle. The sum of its angles at A, B, 
C, cannot exceed two right angles, 
First, let the triangle be Fig. 2. 
nght angled at A. From gy 
Berect BD perpendicular 
to A B, and join CD. 
“a the angles at B 
and C together to exceed a 
right angle. Then because 
ABC and BCA ex- 
ceed a right angle, but A B : 
Cand CBD together only equal a right angle, A C B, contained 
by the sides A C, CB, is greater than CBD contained by the 
sides D B, BC, equal to the others each to each; and, of these, B C 
18 the greater because it subtends the right angle at A (19.1. 
Cor.). Therefore (24.1. Cor.) the angle BDC is greater than 
the right angle at A. And because the three sides C A, A B, 
»U; containing the right angles A, B, are equal to the same, taken 
mm the order DB, BA, AC (4.1. Cor.), and the right angles 
equal in the order B, A, the angle A CD equals the angle 
BDO, and is therefore greater than a right angle. And in like 
Manner may it be shown by taking, in A D produced to R, the 
bases BG, G I, IR, to any desired number, each equal to A B, 
and erecting the perpendiculars GH, IJ, R Y, and so on, 
each equal to AC or BD, that the figures BH, GJ, IY, and 
SO on, are each equal and alike in every respect to the figure A D. 
Complete the figures by joining DH, HJ, J Y, and so on. 
Produce CD indefinitely to K.* Because C DB is greater 
point M, making I M less than IJ or its equal AC. And in 
* It cannot meet A B produced (17.1). 
