268 Mr. B. A. Murray on a rigorous Demonstration 



of the smaller. The extremity A of A B cannot be at A', nor 

 at a point D beyond A' ; for the other extremity B would be 

 beyond B', and then the hypothenuse A' B or D B would be 

 longer than A' B', contrary to the supposition ; it must then be 

 between A' and C, and the other extremity B of the hypothenuse 

 A B must be beyond B'. It is evident, then, that the second acute 

 angle B of A B C is smaller than the second acute angle B' of 

 A' B' C, and that the sides A C, B' C opposite the smaller angles 

 AB C, B' A'C are shorter than the sides A' C, B C opposite the 

 greater angles A' B' C, B A C. 



Second Principal Proposition. 



In a quadrangle, if two opposite sides are equal, and two adja- 

 cent angles comprised by them and a third side are equal and 

 acute, the other two angles are equal and obtuse. 



In the quadrangle ABCD (fig. 10), let the sides AB, C D 



be equal, and the angles B A C, A C D be equal and acute, and 

 suppose A B D, B D C to be right angles. From A and C draw 

 to the opposite sides CD, AB perpendiculars A E, C F, and 

 prolong (either) AE from E until the prolongation E G is equal 

 to AE. Prolong BD from D until the prolongation D H is 

 equal to B D ; draw G H. The quadrangles A E D B, G E 1) H 

 are equal (a), and the quadrangle A B G H is similar to ABCD 

 so far as having two opposite sides A B, G H equal, comprising 

 two equal and acute angles B AE, H GE, and two right angles 

 A B D, G H D,— the other two sides, however, AG, B H, being 

 longer, and the acute angles B A E, H G E smaller than their 

 respective similars. From A and G draw to the opposite sides 



