Proposition 227 Theorem If two lines are cut by three parallel planes, the corresponding segments are proportional. Hypothesis. The lines AB and CD are cut by the || planes MN, PQ, and RS in the points A, E, and B and C, F, and D respectively. Proof. Draw AD, intersecting PQ at G. Let the plane determined by AB and AD intersect PQ in the line EG and RS in the line BD. COR. If two lines drawn from the same point are cut by two or more parallel planes, the corresponding segments are proportional. Ex. 993. Given two parallel planes and a line perpendicular to one of them cutting them at A and B. A third plane parallel to the first two bisects AB; prove that it also bisects the line joining any point of the first plane to any point of the second. DIHEDRAL ANGLES A dihedral angle is the opening between two intersecting planes. The line of intersection is called the edge of the dihedral angle, and the two planes are called its faces. A dihedral angle may be designated by four letters, two on the edge, and one other in each face, the letters on the edge being placed between the 4 other two. For example, the planes AC and AE, intersecting in the line AB, form the dihedral angle CABF. When one dihedral angle stands alone, it may be designated by two letters on the edge; thus, the dihedral angle in the figure above may be designated by AB. B C E The size of a dihedral angle is entirely independent of the extent of the faces. The best way to consider the size of a dihedral angle is to estimate the amount of rotation about the edge which is necessary in order to make one face coincide with the other; the greater the amount of rotation, the larger the angle. Two dihedral angles are equal when they can be placed so that their faces coincide. Two dihedral angles are said to be adjacent when they have the same edge and a common face between them. Two dihedral angles are said to be vertical when they have the same edge and the faces of one are the prolongations of the faces of the other. R M The terms acute, obtuse, oblique, complementary, supplementary, alternate interior, corresponding, etc. can be applied to dihedral angles. The definitions are similar to those for the corresponding cases of angles in Plane Geometry. When one plane meets another plane so as to form two equal adjacent dihedral angles, each angle is called a right dihedral angle, and the planes are said to be perpendicular to each other. For example, the dihedral angles PQRM and PQRN are cqual, and each is a right dihedral angle; the planes MN and PQ are perpendicular to each other. The angle formed by two straight lines, one in each face of a dihedral angle, perpendicular to the edge at the same point, is called the plane angle of the dihedral angle. Proposition 228 Theorem N HINT. Consult Prop. 32 and Prop. 222. Proposition 229 Theorem Two dihedral angles are equal when their plane angles are equal. Hypothesis. The equal & EFG and E'F'G' are respectively plane angles of the dihedral ▲ AB and A'B'. Conclusion. Dihedral AB = dihedral ▲ A'B'. Proof. Apply the dihedral AB to the dihedral ▲ A'B' so that EFG shall coincide with E'F'G', FE falling along F'E', and FG along F'G'. The plane determined by FE and FG coincides with the plane determined by F'E' and F'G'. (?) AB is to the plane determined by FE and FG, and A'B' is to the plane determined by F'E' and F'G'. (?) Then AB coincides with A'B'. (?) .. planes AC and BD coincide with A'C' and B'D' respectively. (?) (?) Q. E. D. COR. I. Plane angles of equal dihedral angles are equal. COR. II. If two planes intersect each other, the vertical dihedral angles are equal. Ex. 994. The intersections of the faces of a dihedral angle with any plane perpendicular to the edge form a plane angle of the dihedral angle. Proposition 230 Theorem Two dihedral angles are in the same ratio as their plane angles. CASE I. When the plane angles are commensurable. Hypothesis. The commensurable CBD and C'B'D' are respectively the plane of the dihedral CABD and C'A'B'D'. Conclusion. Dihedral C'A'B'D' Z C'B'D' Proof. Suppose a Suppose a common measure of CBD and Z C'B'D' to be contained in CBD m times and in C'B'D' n times. Through the several lines of division and the edges pass planes. These planes divide dihedral CABD into m parts and dihedral ▲ C'A'B'D' into n parts, all of which are equal. (?) dihedral C'A'B'D' ZC'B'D' = ་ |