56 



INTRODUCTION. 



cond are perpendicular to it, and therefore parallel to each other 

 (163). 



165. Theorem. If the legs of two angles, not in the 

 same plane, are parallel, the angles are equal. 



B Let AB||CD, and BE|iDF, then ZABE= 



A|^^j^E CDF. Take AB=:BE=zCD=:DF: thenAC|| = 



^][~fa^ BD||=zEF(109), and AEziCF (109); therefore 

 o^^^^^yp ABE and CDF are equal and equiangular. 



166. Problem. To draw a line, perpendicular to a 

 plane, from a given point above it. 



From the point A let fall on any line BC in the 

 given plane a perpendicular AD ; draw DE per- 

 pendicular to BC in the same plane, and from A 

 draw AE perpendicular to DE : then AE will 

 be perpendicular to the plane^BEC ; for if EF 

 be parallel to BC, it will be perpendicular to the plane ADE (163), 

 and consequently to AE ; therefore AE, being- perpendicular to DE 

 and EF, will be perpendicular to the plane passing through them. 



167. Problem. From a given point in a plane, to 

 erect a perpendicular to the plane. 



From any point above the plane let fall a perpendicular on it, and 

 draw a line parallel to this from the given point: this line will be the 

 perpendicular required. 



168. Theorem. If two parallel planes are cut by any 

 third plane, their sections are parallel lines. 



For if the lines are not parallel, they must meet; and, if they meet, 

 the planes in which they are situated must meet, contrarily to the 

 delinition of parallel planes. 



169. Definition. A parallelepiped is a solid con- 

 tained by six planes, three of which are parallel to the 

 other three. 



170. Theorem. The opposite planes of every paral- 

 lelepiped are equal and equiangular parallelograms. 



