OF SPACE. 



35 



163» Theorem. Three straight Mncs, which meet m 

 one point, and are perpendicular to one line, are in one 

 plane. 



Let AB, AC, and AD meet in A, and be per- 

 pendicular to AE, then they are all in one plane. 

 For if either 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 

 angle (161), and EAF=:EAC, the greater to the less : which is im- 

 possible. 



163. Theorem. Two straight lines, which are per- 

 pendicular to the same plane, are parallel to each other ; 

 and two parallel lines are always perpendicular to the 

 same planes. 



Let AB, CD, be perpendicular to the plane 

 BED: draw DE at right angles to BD, and 

 equal to AB, then tlie hypotenuses AD, BE, 

 will be equal, and the triangles ABE, EDA, 

 having all their sides equal, will be equiangu- 

 lar, and the angle ADE will be a right angle: 

 consequently DE is perpendicular to the plane 

 BC (161), and to DC (162), and AB is in the same plane with DC : 

 and ABD and BDC being right angles, AB || CD. 



Again, if AB 1 1 CD, and AB is perpendicular to the plane BED, 

 the triangles ABE and EDA being equiangular, ADE is a right 

 angle: therefore CDE is a right angle (161); but CDB is a right 

 angle (105), 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 perpen- 

 diculars to the two first, and let a plane pass 

 through these perpendiculars : then the third line 

 is perpendicular to this plane (161); consequently tiie first and se- 



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