1842.] On the Theory of Angular Geometry. 237 



Proposition VI. (Fig. 7. J 



The three angles of a triangle are together equal to two right angles. 



Let A B C be the triangle, produce A B, AC to E and F and the 

 base B C both ways to D and G. Then since the lines D B, E B, F C, 

 G C all tend towards the same parts, the angular space D E F G = 

 DE + EF + FG. But D E is the angle D B E or its vertically op- 

 posite A B C ; E F corresponds to the angle B A C and G F is the angle 

 G C F or A C B. Also D E F G is the angular space contained by 

 two portions of the same straight line, it is therefore two right angles. 

 Hence 



ABC + B A C + B C A = 2 right angles. 



Cor. 1 . — The exterior angle is equal to the two interior and opposite 



on the same side, proved by reversing the process of Euclid in the 32. 1. 



or as well thus (see Fig. 7-) The angular space E G is equal to E F 



andFG:EG = EBG;EF = BACandFG = FCG = ACB 



.-. GBE = BAC + ACB. 



Cor. 2. — Euc. I. 16 and 17 are further contained in the last corollary. 



Proposition VII. (Fig. S.J 



The interior angle of a polygon of n sides are together equal to 

 (2 n — 4) right angles. 



Let A B C D E F be the polygon ; subdivide it into triangles by lines 

 from one of the points A. Then the angles of the polygon are equal to 

 the angles of the triangle taken together. Each of the polygon, save 

 the two meeting in A, corresponds to one of these triangles, therefore 

 the number of triangles, is n — 2. And the sum of the angles in each is 

 2 right angles, .\ the sum of all the angles is (n — 2) X 2 right angles. 

 That is, (2 n — 4) right angles. Hence the angles of the polygon are 

 equal to (2 n — 4) right angles. 



Proposition VIII. (Fig. 9.) 



The exterior angles of a polygon, whatever be the number of sides, 

 are together equal to 4 right angles. 



The whole angular space F G H K L F is composed of the angular 



spaces F G, G H, H K, K L, L F. But the whole space F G H K L F 



is the entire angular space on both sides of the line F E, i. e. 4 right 



angles, and each of the constituent angular spaces corresponds to an 



2 i 



