18 



OF SPACE. 



of their intersection, is at right angles to the 

 plane passing through those lines. 



Let AB be perpendicular to CD 

 and EF intersecting each other 

 in A : take AC at pleasure and 

 make ACzrAD=AE=AF i draw 

 through A any line GH, and join 

 ' CEi, DF; then the triangles ADH, 

 ACG are equal and equiangular, 

 AH=:AGandDH=CG;butsince 

 he triangles CBE, DBF, are equal, and equiangular, the 

 angles BCG and BDH are equal, and the triangle BCGz: 

 BDH, BG:=BH, and the triangles ABG, ABH, are equal 

 and equiangular : consequently the angle BAG— BAH, and 

 both are right angles : and the same may be proved of any 

 other line passing through A ; therefore AB is perpendicular 

 to the plane passing through CD and EF (145). 



162. Theorem. Three straight lines 

 Tvhich meet in one point and are perpendi- 

 cular to one line, are in one plane. 



Let AB, AC, and AD meet in A, 

 and be perpendicular to AE, then 

 they are all in one plane. For if ei- 

 ther of them AC is out of the plane 

 which passes through the other two, 

 let a plane pass through AE and AC, 

 and let it cut the plane of AB and AD in AF then the angle 

 EAF is a right ang e (I61), and EAr=EAC, the greater to 

 .the less ; which is impossible. 



163. Theorem. Two straight lines which 

 are perpendicular to the same plane, are pa- 

 rallel to each other ; and two [)arallel lines 

 are always perpendicular to the same planes. 



LetAB, CD, be perpendicular to 

 the plane BED : draw DE at right 

 angles to BD, and equal to AB, then 

 the hypotenuses AD, BE, will be 

 equal, and the triangles ABE, EDA, 

 having all their sides equal, will be 

 equiangular, and the angle ADE will 

 be aright angle: consequent y DE is perpendicular to the 

 plane BC (ISi , and to DC (162>, and AB is in the same 

 plane with DC : and ABD and BDC being right angles, 

 ABIICD. 



Again, if AB ||CD, and AB is perpendicular to the plane 

 BED, the triangles ABE and EDA being equiangular, ADE 

 is a right ang e . tlerefore CDEis a right angle ^l6l) ; but 

 CDB is aii^iiiaJigle (los), therefore CD is perpendicular to 

 BED. 



164. Theorem, Straight lines which are 

 parallel to the same straight line, not in the 

 same plane, are parallel to each other. 



From any point in the third line, 

 draw perpendiculars to the two tirst, 

 and let a plane pass through these per- 



y 



pendiculars : then the third line is perpendicular to this 

 plane (161) ; consequently the first and second are perpen- 

 dicular toil, and therefore parallel to each other (1(13). 



165. Theorem. If the legs of two 

 angles not in the same plane are parallel, the 

 angles are equal. 



Let AB 1 1 CD, and BE II DF, then z. B 



ABE=CDF. TakeAB=:BE=CD=:DF: Ki 

 then AC||=:BD1|=:EF(109), and AE 

 =:CF (log) ; therefore ABE and CDF ^^ 

 are equal and equiangular- 



166. Problem. To draw a line perpen- 

 dicular to a plane from a given point above 

 it. 



From the point A let fall on any line -^i 

 BC in the given plane a perpendicular 

 AD ; draw DE perpendicular 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 per- 

 pendicular 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 definition of parallel planes. 



169. Definition. A parallelepiped is 

 a solid contained by six planes, three of which 

 are parallel to the other three. 



170. Theorem. The opposite planes of 



