20 INTRODUCTION. 



94. Theorem. The angles, which any right line makes 

 on one side of another, are, together, equal to two right 

 angles. 



A p Let AB be perpendicular to CD, and EB obliqne 



/ to it, then CBE+EBD=CBA+ABE-f EBDzz 



CBA+ABD(14). 



C B J) 



95. Theorem. If two right lines make with a third, 

 at the same point, but on opposite sides, angles together 

 equal to two right angles, they are in the same right line. 



-J) If it be denied, let AB, which together with AC, 

 /^g makes with AD, the angles BAD, DAC equal to two 



P a"^^""!^ ^'o^i* angles, be not in the right line CAE. Then 



BAD+DAC, being equal to two right angles, is 

 equal to EAD-fDAC (94), and BAD=zEAD, the less to the greater, 

 which is impossible. 



96. Theorem. If two right lines intersect each other, 

 the opposite angles are equal. 



From the equals, ABC+ABD and ABD+DBE 

 (94, 82), subtract ABD, and the remainders, ABC, 



DBE, are equal. In the same manner ABDz= 

 ^ CBE. 



97. Theorem. If one side of a triangle be produced, 

 the exterior angle will be greater than either of the interior 

 opposite angles, 



A E bisect AB in C, draw DCE; take CEzzCD, 



and join BE, then the triangle ACD— BCE (96, 



f\ / S6), and ZCBEizCAD; but ABF>CBE, tliere- 



j) j3 :, fore ABF>CAD. And in the same manner it 



may be proved, by producing AB, that^ABF is 

 greater than AD B. 



98. Theorem. The greater side of any triangle is 

 opposite to the greater angle. 



