THEOREM XXIV.

62. Triedral angles whose face angles are equal and similarly situated are equal.

whose vertex is E; and let the faces in each be arranged in the same order; then the triedral angles are equal.

On the edges of the triedral angles cut off the six equal distances A B, AC, AD, EF, EG, EH; through the points B, C, D pass a plane (2) whose intersections with the faces of the triedral angle A are BC, CD, DB; also through the points F, G, H pass a plane whose intersections with the faces of the triedral angle E are FG, GH, HF. The isosceles triangle BAC FEG (I. 80), and hence B CFG; in like manner CD= GH, and BD=FH; hence the triangles BCD, FGH, being mutually equilateral, are equal (I. 88), and the angle DBC=HFG. At any point I in AB, draw IK in the face BA C, and IL in the face B A D, perpendicular to A B; as BAC, BAD are isosceles triangles, the angles A B C, AB D, are acute; hence I K, IL will meet B C, B D respectively; let K and I be the points of meeting; join KL. From EF cut off EM: AI and construct MNO in the same manner as I KL was constructed.

Now as ABEF and AI EM, IBMF; the angle ABC EFG, and the right triangles BIK, FM N are equal (I. 81) and I K= MN and BK=FN. In like manner it can be proved that ILM O and B LF 0. Hence the

triangle B KL⇒ F N O (I. 80). Now the triangles IK L, MN O, being mutually equilateral are equal (I. 88), and the angle LIK, the measure (34) of the diedral angle whose edge is AB is equal to D M N, the measure of the diedral angle whose edge is E F Now if the triedral angle A is placed on E so that the face B A C coincides with its equal FE G, as the diedral angles whose edges are AB, EF are equal; the face BAD will lie on its equal FEH, and AD on EH; hence DAC will coincide with HEG (2), and the triedral angle A with the triedral angle E.

a b and eg can be proved equal in the same manner as in (61); and also the diedral angles whose edges are ac and ef; but the two figures cannot be made to coincide. In this case the two triedral angles are said to be equal by symmetry, or symmetrically equal, or simply symmetrical.

Good examples of two symmetrical objects are the right and left hand glove which can be made of exactly equal pieces but arranged in reverse order. The right and left hands are also good examples; or an object and its apparent image in a mirror.

64. Scholium 2. If each of the face angles BAC, BAD (and also their equals FEG, FEH) is acute, the proof is made very simple by passing planes through B and F perpendicular to the edges A B, EF respectively. Then the angles DBC, HFG are respectively the measures of the diedral angles whose edges are A B, E F. This method cannot be used when the angle BA C, e. g. is not acute, since the perpendicular plane through B would not cut the edge A C.

EXERCISES.

65. If several planes intersect another plane in the same straight line,

1st. The sum of all the diedral angles on one side of this plane is equal to two right diedral angles. (34, 35; I. 44.)

2d. The sum of all the diedral angles thus formed is equal to four right diedral angles. (I. 46.)

66. If two planes intersect one another,

1st. Each diedral angle is the supplement of its adjacent diedral angle. (I. 45.)

2d. The vertical diedral angles are equal. (I. 48.)

67. In like manner by changing the word point to straight line, straight line to plane, inserting diedral before the word angle, etc., state and prove, as propositions relating to planes, 47, 49, 54, 55, 56, of Book I.

68. If the face angles that form the polyedral angle A in the Fig. in Art. 60 become equal to four right angles, what then?

69. Two triedral angles are equal, or symmetrical,

1st. If they have two face angles, and the included diedral angle of the one equal respectively to two face angles and the included diedral angle of the other. (I. 80.)

2d. If they have two diedral angles and the included face angle of the one equal respectively to two diedral angles and the included face angle of the other.

3d. If the three diedral angles of the one are equal respectively to the three diedral angles of the other.

70. Is there anything in plane triangles analogous to (69, 3d)?

71. State with the necessary changes to make the application to triedrals, 82, 83, 84, 85, 86, 87, 101, 102, of Book I. Prove each proposition.

72. Show that two triedral angles may have three parts of which one is a side, or even four of which two are sides, respectively equal, and be neither equal nor symmetrical.

73. The planes bisecting the diedral angles of a triedral angle intersect in the same straight line; and any point of this line is equally distant from each face of the triedral.

74. The planes passing through the lines that bisect the face angles of a triedral angle, and perpendicular to the faces respectively, intersect in the same straight line; and any point of this line is equally distant from the edges of the triedral.

75. The planes passing through the edges of a triedral angle, and perpendicular to the opposite faces respectively, intersect in the same straight line.

76. The planes passing through the edges and the lines bisecting the face angles of a triedral angle respectively, intersect in the same straight line.

77. If one face of a triedral angle is rectangular, an adjacent diedral angle and its opposite face are both acute, or both right, or both obtuse.

78. If from a point within a triedral angle perpendiculars are drawn to the three faces, these perpendiculars will be the edges of a triedral angle whose face angles will be supplements respectively of the measures of the diedral angles of the given triedral angle. (39, 38; I. 125.)

79. Corollary. The face angles of the first triedral angle named in (78) are also supplements of the measures of the diedral angles of the second triedral angle. These triedral angles are called supplementary triedral angles of each other.

80. The sum of the diedral angles of a triedral angle is greater than two, and less than six right diedral angles. (60, 79.)

81. If the diedral angles of a triedral angle become equal to two right diedral angles, what then? what, if equal to six?

BOOK VII.

POLYEDRONS.

DEFINITIONS.

1. A Polyedron is a solid bounded by planes.

The bounding planes are called faces; their intersections, edges; the intersections of the edges, vertices.

2. A polyedron with four faces is called a tetraedron; with six, a hexaedron; with eight, an octaedron; with twelve, a dodecaedron; with twenty, an icosaedron.

3. The Volume of a solid is the measure of its magnitude. It is expressed in units which represent the number of times it contains the cubical unit taken as a standard.

4. Equivalent Solids are those which are equal in volume.

PRISMS AND CYLINDERS.

5. A Prism is a polyedron two of whose faces are equal polygons having their homologous sides parallel, and whose other faces are parallelograms. Corollary. The lateral edges are equal to each other.

The equal parallel polygons are called bases; as A B and CD; and the other faces together form the convex surface.

6. The Altitude of a prism is the perpendicular distance between its bases; as E F.