334 A. CO. Twining—Euelid’s Doctrine of Parallels. 
Prop. XIX. Cor. In a right angled triangle the side sub- 
tending the right angle is greater than either of the sides con- 
taining the right angle. 
For (17. 1.) the right angle must be greater than either of the 
angles opposite to it, and therefore must subtend a greater side. 
Prop. XXIV. Cor. In two triangles having unequal angles 
contained by sides equal each to each, the angle opposite the 
smaller contained angle and subtended by that one of the con- 
taining sides which is not less than the other, is greater than the 
similarly subtended side of the other triangle. 
For DEF is greater than DEG or A BC, which has the 
larger included angle. 
Prop. XXI a. THEOREM. 
If two straight lines intersect, then any third line from one to 
the other is greater than a perpendicular dropped from any point 
between the third line and the point of intersection. 
Let the straight lines A E, A F, intersect in A, and let BC 
be perpendicular to A F, and A E be longer than AB, Then 
C 
any line E D is longer than BC. 
For suppose ED not Fig. 1. 
i 
terior opposite angle E AG, which is impossible. Therefore 
EF can neither be less than BC nor equal to it, but must fi 
greater; and much more must any other line ED from E be 
greater. 
