132 PROCEEDINGS OF THE AMERICAN ACADEMY 



as to make the angles DC F, G FH each equal to the angle BAG, 

 and connect the points D and jP, D and G, D and j5, by right lines ; 

 also through D draw D K, at right angles to D G. Since AB = D C, 

 A C = CF, and the angle BAG equals the angle D C F, the trian- 

 gles ABC, G D F arc identical (Sim., B. I., p. 4), making the sides 

 I) F and B G equal to each other, and the angle G DF equal to the 

 angle A B G, and the angle G F D equal to the angle A G B ; hence 

 the sum of the angles BAG, B G A equals the sum of the angles 

 B GA, D GF, which equals the sum of the angles G F D, G F H. 

 Since the sum of the three angles at G makes two right angles, and 

 that the sum of the angles at F makes two right angles (Sim., B. I., 

 p. 13), it follows from what has been proved that the angles B G D, 

 DFG are equal, and since BG^=DF, G D^^^ F G, the triangles 

 BGD, DFG are identical (Sim., B. I., p. 4), making BB equal to 

 D G, and the angle G B D equal to the angle F D G, and the angle 

 G DB equal to the angle F G D ; hence the sum of the angles G B D, 

 GDB equals the sum of the angles G D B, F D G. By Prop. 2, 

 since the sum of the three angles of any triangle is not greater than 

 two right angles, and that the three angles at G make two right angles, 

 it follows that the sum of the angles B and D of the triangle BCD 

 is not greater than the suni of the angles ACB, F G D ; hence, and 

 from what has been proved, it follows that the sum of the angles of 

 the triangle AB G is not less than the sum of the angles B DC, CDF, 

 F D G ; but it is evident that the sum of these angles exceeds the 

 right angle K D G hy a. fine angle ; hence the sum of the three an- 

 gles of the triangle ABC exceeds a right angle, as required. 



" Remark. If A B is extended in the direction A B to E, so that 

 B E^=- B D, and if the points D and E, G and E, are connected by 

 right lines, the point D falls evidently within the pentagonal figure 

 G B E G F', and if jR denotes a right angle, the sum of the angles 

 BDG, CDF, FDG, G D E, EDB is equal to 4 E (Sim., B. I., 

 p. 15, Cor. 2). Since the sum of the angles AB D, DB E is equal to 

 2 R, the sum of the angles at the base of the isosceles triangle B D E 

 is not greater than the angle A B D ; consequently the angle B DE 

 is not greater than the angle AB D -^^, which is not greater than half 

 the sum of the angles B D C, CDF, F D G. Wc now observe that the 

 sum of the angles of the triangle A B C is not less than R -\- /'g R. 

 For if the sum of the angles of the triangle A B G is not greater than 

 R -j- ■f'j; R, then by what has been shown the sum of the angles BDG, 



