SECTION IV. ANGLES BETWEEN PLANES

64. Dihedral Angles. An angle between two intersecting planes is called a dihedral angle. The planes are called the faces, and their intersection is called the edge of the dihedral angle.

Two intersecting planes evidently form four dihedral angles, just as two intersecting lines form four plane angles. In the figure, the planes M and N, intersecting in XY, form the dihedral angles indicated by the arrows as A, B, C, and D. A dihedral angle may be denoted by naming its faces, with or without the use of the intersection, as dihedral angle MN, or dihedral angle M-XY-N. When, as in this figure, there might arise confusion as to which angle was intended, a second letter may be used for each plane, as M' and N' in the figure, and their order in the naming of the angle will determine which angle is meant. Thus, angle C could be named M'-XY-N', the direction of rotation being considered, as in plane geometry, counterclockwise (that is, in the direction opposite to that in which the hands of a clock rotate).

As the size of a plane angle does not depend on the length of its arms, so the size of a dihedral angle does not depend on the extent of its faces, but only on the amount of rotation necessary for a plane to rotate about the intersection from one face to the other.

Familiar illustrations of dihedral angles are, the angle between two leaves of an open book, the angle between two walls of a room, the angle between two planes of a roof, the various angles made as a desk cover is raised or as a door is opened, etc.

65. Plane, or Measuring, Angle. If at a point in the edge of a dihedral angle, two perpendiculars, one in each face, are drawn to the edge, the angle between the perpendiculars is called the plane angle, or the measuring angle of the dihedral angle. This plane angle evidently lies in, and its arms determine, a plane perpendicular to the edge. It is evident that all plane angles of the same dihedral angle are equal. Why?

66. Theorem XVII. angles are equal.

Superpose.

Plane angles of equal dihedral

37. The greater of two unequal dihedral angles has the greater plane angle.

67. Theorem XVIII. Two dihedral angles are equal if their plane angles are equal.

38. Vertical dihedral angles are equal. (See def. in § 70.)

68. Theorem XIX. Two dihedral angles are proportional to their plane angles.

Assume a common divisor for the commensurable case.

69. Measurement of a Dihedral Angle. Since the amount of rotation in a dihedral angle is the same part of a complete rotation about its edge that its plane angle is of a perigon (Th. XIX), the plane angle is the measure of the dihedral angle, this complete rotation being considered as a standard by which to measure. It is on this account that the plane angle is called the measuring angle of the dihedral angle.

It is evident, therefore, that propositions concerning dihedral angles will usually be proved by the use of their measuring angles.

70. Terms used for Plane Angles that apply also to Dihedral Angles. On account of the dependence of dihedral angles on their plane angles, a dihedral angle is called acute, right, obtuse, straight, or reflex, according as its plane angle is acute, right, obtuse, straight, or reflex; also two dihedral angles are adjacent, vertical, complementary, supplementary, or explementary, according as their measuring angles fulfill the conditions of the definitions of these terms.

If the dihedral angle between two planes is a right dihedral angle, the planes are said to be perpendicular to each other.

71. Theorem XX. If a line is perpendicular to a plane, any plane through that line is perpendicular to the given plane.

How can planes be proved perpendicular? What auxiliary line does this necessitate?

Note that this is a second method of proving planes perpendicular, and that it is simpler to apply than the definition, since that requires the use of two lines, while this method requires but one.

39. How many intersecting planes can be perpendicular to the same plane?

72. Theorem XXI. A line drawn in one of two perpendicular planes, perpendicular to their intersection, is perpendicular to the other plane.

This proves that each of two perpendicular planes contains an unlimited number of lines perpendicular to the other plane. Are all the lines in one plane perpendicular to the other? Are all the lines that meet the intersection perpendicular to the other plane?

40. If a line and a plane intersect, a plane perpendicular to one is not perpendicular to the other.

73. COR. 1. If two planes are perpendicular, a line perpendicular to one of them through a point in the other, lies wholly in that other.

Use the theorem.

Stated like the other methods of proving a line in a plane, this becomes: A line is in a plane if it has one point in that plane, and if the line and the plane are both perpendicular to the same plane.

74. COR. 2. If two intersecting planes are each perpendicular to a third plane, their intersection is perpendicular to that plane.

What is known of a perpendicular to the third plane through any point of the given intersection?

41. Can planes perpendicular to intersecting planes be parallel? Explain.

75. COR. 3. If two intersecting lines are perpendicular to the faces of a dihedral angle, they determine a plane that contains one of the measuring angles of that dihedral angle.

76. Theorem XXII. Through any line not perpendicular to a given plane, one, and but one, plane can be passed perpendicular to that plane.

Use the simplest method of determining a plane perpendicular to the given plane, then show that a perpendicular plane must contain these determining lines.

77. Projections. The projection of a point upon a plane is the foot of the perpendicular from that point to the plane. The projection of a line upon a plane is the locus of the projections of all its points upon that plane.

78. Theorem XXIII. The projection upon a plane of a line that is not perpendicular to that plane is a straight line.

Prove (1) the projections of all its points will lie in one line; (2) any point in that line is the projection of some point of the given line. What would be the projection of a perpendicular upon a plane?

NOTE. The following group of exercises on projections is closely associated with the elements of descriptive geometry.

Given two planes perpendicular to each other:

42. Find the line of which two given lines, one in each plane (but not both perpendicular to the intersection) are the projections upon those planes.

43. If a line is perpendicular to one of the planes, its projection upon the other is perpendicular to the intersection.

44. If a line is parallel to one of the planes, its projection upon the other plane is parallel to the intersection.

45. If a line is parallel to both planes, its projections upon them are parallel to each other.

46. If two intersecting lines are projected upon both planes, the line joining the intersection of their projections upon one plane with