22 INTRODUCTION. 



to CD, let it be the greater, and let AGiinCD ; then, by what has 

 been demonstrated, the triangle AFGzzCED, and ZAGF— CDE=: 

 ABF, by the supposition ; but AGF> ABF (97), which is impossible. 



103. Theorem. The shortest of all right lines, that 

 can be drawn from a given point to a given right line, is 

 that which is perpendicular to the line, and others are 

 shorter as they are nearer to it. 



Let AB be perpendicular to CD, then AB is 

 shorter than AD. Produce AB, take BEizAB, 

 and join DE; then the triangle ABDzzEBD (86), 

 and ADziDE. But AB+BE or 2AB is less than 

 AD+DE or 2AD (79), therefore AB<AD (18). 

 In a similar manner 2AD<2AF (80), and AD < 

 AF. 



104. Theorem. If a right line, cutting two others, 

 makes the alternate angles equal, the two lines are pa- 

 rallel. 



y' If Z. ABC=:ADE ; BC and DE are 



—p^ ~^^ parallel ; for if they meet, as in F, they 



~~75 E will form a triangle BDF, and Z.ADE 



^ >ABC (97). 



105. Theorem. A right line, cutting two parallel lines, 

 makes equal angles with them. 



Let AB cut the parallels BC, DE ; tlien if 



/.ABC is not equal to ADE, let it be equal to 



ADF, then BC and DF are parallel (104), and 



■ ^Yy — j£ DE, which cuts DF, will also, if produced, cut 



^ BC (83), contrarily to the supposition. 



106. Theorem. Right lines, parallel to the same line, 

 are parallel to each other. 



Let AB and CD be parallel to EF ; draw 

 GHI cutting them all, then /KGBziKIF 

 (105), and /.KHDzzKIF, therefore Z.KGB 

 =KHD, and AB 1 1 CD (104). 



