GEOMETRY. 



Let the line AB be parallel to the line CD 

 situated in the given plane 

 P ; it shall be parallel to the 

 plane itself. The plane of 

 the two parallels cuts the 

 given plane P along the line 

 CD ; hence if AB intersects 

 the plane P, it must be on 

 the line CD; but AB and 

 CD never intersect, since 

 they are parallel ; hence AB and the given plane 

 P are parallel. 



Conversely, if AB be parallel to the plane P, 

 then the intersection CD of a plane through AB 

 and the given plane will be parallel to AB. 



PROP. X. If a straight line be drawn parallel 

 to a plane, the portions of parallel lines intercepted 

 between the line and the plane 

 are equal. 



Let the line AB and the 

 plane P be parallel to each 

 other, and let AC and BD be 

 parallel, then AC and BD are 

 equal. The plane of AC and 

 BD cuts the given plane P 

 along the line CD parallel to AB ; therefore the 

 figure ABDC is a parallelogram, and AC is equal 

 to BD. 



PROP. XI. Two planes which are perpen- 

 dicular to the same straight 

 line are parallel to each \ 

 other. ' . 7 



These planes cannot inter- 

 sect, for only one plane can 

 be drawn through a point per- 

 pendicular to another plane. 



Cor. i. Hence it follows that lines drawn 

 through a point A, parallel 

 to a given plane P, lie in 

 a plane parallel to the given 

 plane. 



Cor. 2. If two planes are 

 parallel, every plane perpen- 

 dicular to the one is also 

 perpendicular to the other. 



PROP. XII. If two straight lines are cut by 

 three parallel planes, they will be cut proportionally. 



Let the three parallel planes 

 P, Q, R cut the two lines 

 ABC, DEF ; to prove that /; ~7 



AB_DE /_ 



BC~EF' 



Join AF, cutting the plane Q 

 in G. Join BG, GE. Then 

 we have (Prop. XXX., p. 622) 



AG _ DE 

 GF " "EF' 



AG AB . 

 GF BC ; 



AB DE 



'* BC ~ EF' 



THE LINE AND PLANE INCLINED TO EACH 

 OTHER. 



Definition. The projection of a point on a 

 plane is the foot of the per- 

 pendicular drawn from the 

 point on the plane. 



The projection of a line 

 upon a plane is the locus of 

 the projections of the differ- 

 ent points of the line on the 

 plane ; thus a is the projec- 

 tion of A, and ab that of AB on the plane P. 



PROP. XIII. The projection of a straight line 

 on a plane is a straight line. 



Let ABC be a straight 

 line, and let a, b, and c be 

 the feet of the perpendicu- 

 lars from any three points 

 A, B, C of the line on the 

 plane P ; then abc will be a 

 straight line. For the plane 

 which contains AC and Aa will (Prop. IV., p. 627) 

 contain all the perpendiculars, and it will inter- 

 sect the plane P in the straight line abc. 



PROP. XIV. If a straight line is oblique to a 

 plane, the angle which it makes with its projection 

 is less than that which it makes with any other 

 line that meets it in that plane. 



Let AB be the given line, meeting the plane P 

 in A. From B draw BC 

 perpendicular to the plane. 

 Then AC is the projection of 

 AB on the plane ; and it is 

 required to prove that the 

 angle BAC is less than BAD, 

 formed by AB and another 

 line AD from A. Make AD 

 equal to AC, and join BD ; then BD (Prop. V., 

 p. 627) is greater than the perpendicular BC. 

 And in the two triangles BAD, BAC we have 

 the sides BA, AD in the one equal to BA, AC hi 

 the other ; but BD is greater than BC, therefore 

 angle BAG is less than BAD. 



Scholium. The inclination of a line to a plane 

 is measured by the angle the line makes with its 

 projection on the plane. 



THE PLANE INCLINED TO THE PLANE. 



Definition. The angle between two planes is 

 called a dihedral angle ; thus the figure formed 

 by the two planes ABC, DEC, 

 which intersect along BC, is a 

 dihedral angle; the line BC is 

 called its edge, and the planes 

 AC and DC its faces. The 

 dihedral angle is named by the 

 two letters which distinguish its 

 edge, if there is only one angle 

 having this line for its edge; 

 otherwise it is named by four 

 letters, of which the two middle ones denote the 

 edge ; thus the dihedral angle ABCD is the angle 

 formed by the two faces AC and DC. 



A dihedral angle is measured by the angle 

 between the two lines, drawn one in each plane 

 perpendicular to the intersection of the planes 

 from any point in that line. Thus the dihedral 



