OF ARTS AND SCIENCES. 



143 



"Let A C B he the triangle, having the angle C right; from C 

 draw CD, at right angles to the hy- 

 pothenuse, meeting it at Z> ; hence 

 cor., prop. 4, since the right tri- 

 angles ACD, ACB have the 

 angle A common, their other acute 

 angles AC D and B are equal ; al- 

 so since the (right) triangles BCD, 



ABC have a common angle, B, their other acute angles BCD 

 and A are equal. Hence the sum of the angles A and B is equal to 

 the sum of the angles BCD and AC D, which compose the right 

 angle ACB, and of course the sum of the angles A and B is equal 

 to a right angle ; and consequently the sum of all the angles of the 

 triangle ACB \s equal to two right angles. Again, the right tri- 

 angles ACB, AC D having the common angle A, by prop. 3, give 

 the equality ^^ = -j^, or AC^=^AB . AD ; and in the same way 

 we get from the triangles A CB, BCD, BC^=AB.BD; and 

 consequently A C^ -\- B C^ =^ A JB^, as required. 



" Cor. Since the right triangles A CD, BCD have the angles 

 CAD, BCD equal, they (by the cor. to prop. 4) give the equality 

 ^ = ^,ovCD^ = AD.BD. 



" Prop. 6. The sum of the angles of any triangle is equal to two 

 right angles. 



" Let ACB denote any triangle ; and suppose that the angle C is 

 not less than either of the other 

 angles of the triangle, and that the 

 perpendicular C D is drawn from 

 C to the opposite side A B ; then, 

 Sim., p. 17, B. I., CD will fall 

 within the triangle ACB. Hence, 

 since the triangles AC D, BCD 



are right-angled at D, by the last prop, the sum of the acute angles 

 A and ^ C i> of the first of these triangles is equal to a right angle ; 

 and in the same way the sum of the acute angles B and B C D of the 

 second triangle is equal to a right angle ; but the sum of the acute 

 angles of these triangles equals the sum of the angles of the triangle 

 ACB; consequently, the sum of the angles of the triangle ACB 

 is equal to two right angles, as required. 



" In conclusion, we will remark that the relation of what has been 



