42 SCIENCE ABSOLUTE OF SPACE. 



the equivalence of the triangles AGC, AGH, 

 standing on the same AG, and having their 

 vertices on the line PQ. Moreover, ACF = 

 CAG, GCQ=CGA, and ACF+ACG+GCQ= 

 st.Z (32); and so also CAG+ACG+CGA= [23] 

 st. Z ; therefore, in any triangle ACG of this 

 sort, the sum of the three angles =st. ^. But 

 whether the straight AG may have fallen upon 

 AG (which III MN), or not; the equivalence of 

 the rectilineal triangles AGC, AGH, as well 

 of themselves, as of the sums of their angles, 

 is evident. 



40. Equivalent triangles ABC, ABD, 

 ^(henceforth rectilineal), hav- 

 ing one side equal, have the 

 sums of their angles equal. 

 For let MN bisect AC and 

 B BC, and take (through C) 



FIG. 36. PQlllMN; the point D will 



fall on line PQ. 



For, if ray BD cuts the straight MN in the 

 point E, and so ( 39) the line PQ at the dis- 

 tance E)F=E}B; we shall have AABC=AABF, 

 and so also AABD=AABF, whence D falls 

 at F. 



But if ray BD has not cut the straight MN, 

 let C be the point, where the perpendicular bi- 

 secting the straight AB cuts the line PQ, and 



