﻿448 DEMOXSTRATIOX OU THE I2TH AXIOM 



Suppose a c to be greater or less tlian d d, take 

 a o~b d and draw do: now, since a o:izb dy It d o 

 will be a right angle (prop. ^) and therefore equd to 

 b d c, wliicli is impossible :■ a c cannot be greater nor 

 less than b d:- a c:::::b d, and therefore (by the fore- 

 going proposition) c d'^za b and a c d u. right angle. 

 QED. 



Prop, 4. 



If two right lines, 

 a n, b 0, perpendicu- 

 lar to the same right 

 line a b, be cut by a 

 ri-iit line r s ; the 

 alternate angles will 

 be equal ; the exter- 

 a b nal aiigle equal to 



the internal remote angle on the same side of the 

 cutting line; and the two internal angles, on the 

 same side, equal to two right angles. 



If the cutting line r s be perpendicular to one of 

 the given lines, it will be perpendicular io tlie other 

 (by the foregoing prop. ) and theielore ali the angles 

 right, and consequently equal. 



If the cutting line r s be not perpendicular, draw 

 the perpendicular c m, d e ; by the former proposi- 

 tion c 7Jizza bzze d ; also the angle m d e a. right an- 

 gle;:- by the 2d prop. ct'i=:w2fl?:- the triangles ced, 

 cmd, are mutually equilateral ; and therefore (8. I.) 

 ecd:=zcdm ; and consequently their complements ncd 

 Si.r\d bdc are equal ; again bds:=irdo~.acs ; again acd 

 -i-bdczzmdc + bdczzto two right angies. Q E D. 



Prop. 



