IV. POLYHEDRAL ANGLES

78. Polyhedral Angle. If three or more planes meet in a point, they form a polyhedral angle.

C

B

If we consider angles greater than 360°, two lines which meet form an infinite number of plane angles; but this fact never confuses us in plane geometry. Similarly, in solid geometry, if three planes meet in a point, they form an infinite number of polyhedral angles; but since we shall always make clear the angle to be considered, this fact will cause no difficulty.

V

A

The three planes AVB, BVC, and CVA, in the above figure, form a polyhedral angle which we designate by V, or by V-ABC. The letters are given in the order in which they occur around the figure.

If the figure formed on a plane which cuts all the planes forming a polyhedral angle is convex, the angle is said to be convex; if the figure is concave, as here shown, the angle is y Isaid to be concave. Since we shall consider

E

E

B'

A'

only convex polyhedral angles, this distinction need not be memorized.

79. Parts of a Polyhedral Angle. The common point at which the planes meet to form a polyhedral angle is called the vertex of the angle. The intersections of the planes are called the edges of the angle. The portions of the planes lying between the edges are called the faces of the angle. The angles formed by adjacent edges are called the face angles of the polyhedral angle. The vertex, edges, faces, face angles, and the dihedral angles formed by the faces are the parts of a polyhedral angle.

In a polyhedral angle of n faces there are n edges, n face angles, and n dihedral angles.

In the first of the above figures, the vertex is V; the edges are VA, VB, VC; the faces are AVB, BVC, CVA; the face angles are ZAVB, ZBVC, LAVC; and the dihedral angles are VA, VB, VC.

80. Classes of Polyhedral Angles. A polyhedral angle of three faces is called a trihedral angle.

Polyhedral angles of four, five, six, seven, tetra-, penta-, hexa-, hepta-,

....

...

faces take the prefixes

Since, however, we rarely refer to polyhedral angles of more than three faces, these names need not be memorized.

V

81. Equal Polyhedral Angles. If two polyhedral angles can be placed so that their vertices and edges coincide, the angles are said to be equal.

Conversely, if polyhedral angles are equal, they can be made to coincide by superposition, because, if the vertices and edges coincide, all the correspond

C

C'

B'

ing parts coincide. In this figure the trihedral V and V' are equal.

Exercises. Review

1. A reading lamp is attached to an upright rod which is fastened to two iron pieces, or feet, resting on the floor as here shown. If the rod is perpendicular to the two pieces, is it perpendicular to the floor? Would three pieces be better? Would four be better? Would five be still better? State the geometric principles involved in your answers.

2. Two adjacent walls and the ceiling of a rectangular room form a trihedral angle. Write

a statement of the relations of the parts of the angle; for example, that each dihedral angle has a certain size.

3. It is known that the projections of four points upon a plane (that is, the feet of the perpendiculars from the points to the plane) lie in a straight line. Write your conclusion as to whether or not these points lie in a straight line; lie in a plane; are scattered at random in space.

82. Symmetric Polyhedral Angles. If the faces of the polyhedral V-ABCD are produced through the vertex V, another polyhedral angle, the V-A'B'C'D', is formed. The ZV-A'B'C'D' is said to be symmetric with respect to ZV-ABCD. The face figure at the left,

AVB, BVC, in the

are equal respec

A'VB', B'VC',

B'

CDDA

B' D'A'

tively to the face

of the polyhedral

V-A'B'C'D' (§ 6). A

Also, the dihedral VA, VB,

...

are

equal respectively to the dihedral

▲ VA', VB', · · · (§ 68, 3). The figure at the right shows a pair of these vertical dihedral angles.

Looked at from the point V, the edges of ZV-ABCD are arranged counterclockwise (from left to right) in the order VA, VB, VC, VD, but the edges of ZV-A'B'C'D' are arranged clockwise (from right to left) in the order VA', VB', VC', VD'; that is, in an order which is the reverse of the order of the edges of V-ABCD. Therefore,

Two symmetric polyhedral angles have all their parts equal, each to each, but arranged in reverse order.

X

B'

V-Y

83. Symmetric Polyhedral Angles not Superposable. In general, two symmetric polyhedral angles cannot be superposed. If the trihedral V-A'B'C' here shown is made to turn 180° about XY, the bisector of ZCVA', then VA' coincides with VC, VC' with VA, and the face A'VC' with the face AVC. But since the dihedral VA, and hence the dihedral VA', is not equal to the dihedral ZVC, the face A'VB' does not coincide with the face BVC, nor does C'VB' with AVB. Hence VB' takes some position as VB"; that is, symmetric trihedral angles cannot, in general, be superposed.

A

C

Proposition 17. Trihedral Angles

84. Theorem. If the three face angles of one trihedral angle are equal respectively to the three face angles of another, the trihedral angles are either equal or symmetric.

AAA

P' Z

Given the trihedral V-XYZ and V'-X'Y'Z' with face 4YVX, ZVX, ZVY equal respectively to face Y'V'X', Z'V'X', Z'V'Y'.

Prove that the trihedral V-XYZ and V'-X'Y'Z' are either equal or symmetric.

Proof. On the edges of the trihedral / take the six equal segments VA, VB, VC, V'A', V'B', V'C'.

From any point P on VA construct PQ in the face XVY and PR in the face XVZ, each to VA.

§ 13, 1

Since

VAB and VAC are equals of isosceles A,

VAB and VAC are acute.

$ 8,4

Hence the Is PQ and PR meet AB and AC respectively.

In the faces X'V'Y' and X'V'Z' construct P'Q' and P'R' respectively, each 1 to V'A'.

Now, since ABAC is congruent to AB'A'C',

Proved

The symmetric angles are shown in the two figures at the right.