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Exercises. Review

1. Make a list of the numbered propositions in Book VI, stating under each the previous propositions in Book VI employed in its proof.

2. Make another list of the numbered propositions, stating under each the subsequent propositions in Book VI in which it is used as an authority.

3. Mention three trihedral angles which may be found in a room, and state whether they are equal, symmetric, or both equal and symmetric.

4. By consulting a dictionary find the derivation of the words "dihedral," "trihedral," and "polyhedral," and explain how these derivations apply to the figures.

5. If each of two trihedral angles have right angles for their face angles, they are both equal and symmetric.

6. Consider whether or not two trihedral angles are equal if two face angles and the included dihedral angle of one are equal respectively to two face angles and the included dihedral angle of the other, and give the proof.

7. State a condition under which two polyhedral angles of four faces are equal, and prove the equality.

8. In the trihedral ZV-ABC, what is the locus of points equidistant from the faces VAB and VBC? from the faces VAB and VCA? from the faces VAB and VBC, and also from the faces VAB and VCA? To what proposition in plane geometry does this correspond ?

9. If two intersecting planes pass through two parallel lines a and b respectively, their line of intersection is parallel to a and b.

10. A line parallel to each of two intersecting planes is parallel to their line of intersection.

BOOK VII OF GEOMETRY

POLYHEDRONS, CYLINDERS, AND CONES

I. PRISMS

85. Polyhedron. A solid bounded by planes is called a polyhedron.

For example, in the second figure of $ 84, if we consider that part of the ZV-XYZ cut off by the plane ABC, we have a polyhedron.

The bounding planes are called the faces of the polyhedron, the intersections of the faces are called the edges of the polyhedron, and the intersections of the edges are called the vertices of the polyhedron.

A line joining any two vertices not in the same face is called a diagonal of the polyhedron.

If every plane which cuts a given polyhedron forms a convex polygon, the polyhedron is said to be convex. We shall consider only convex polyhedrons in this course.

86. Prism. A polyhedron of which two faces are congruent polygons in parallel planes, and the other parallelograms having two of their sides in the two parallel planes, is called a prism.

The figure at the right shows a prism. The parallel polygons are called the bases of the prism, the parallelograms are called the lateral faces, and the intersections of the lateral faces are called the lateral edges. The lateral edges of a prism are equal ($ 10, 3).

The sum of the areas of the lateral faces is called the lateral area of the prism.

The perpendicular distance between the planes of the bases is called the height or altitude of the prism.

ces are

87. Right Prism. A prism whose lateral edges are perpendicular to its bases is called a right prism.

The first of the figures below shows a right prism. The lateral edges of a right prism are equal to the altitude ($ 58).

88. Oblique Prism. A prism whose lateral edges are oblique to its bases is called an oblique prism.

The second of the figures below shows an oblique prism.

[graphic]

89. Prisms classified as to Bases. Prisms are said to be triangular, quadrangular, and so on, according as their bases are triangles, quadrilaterals, and so on.

Thus, the first figure above shows a quadrangular prism, the second shows a triangular prism, and so on.

90. Right Section. The polygon formed by the intersections of the lateral faces of a prism with a plane which cuts all the lateral edges, produced if necessary, and is perpendicular to them, is called a right section.

The third figure above shows how a right section is formed.

91. Truncated Prism. The part of a prism included between the base and a section made by a plane oblique to the base is called a truncated prism.

The fourth of the above figures shows a truncated prism.

$8 87-92

SECTIONS OF A PRISM

59

Proposition 1. Sections of a Prism 92. Theorem. The sections of a prism made by parallel planes cutting all the lateral edges are congruent polygons.

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Given the prism PR and the sections AD and A'D' made by Il planes cutting all the lateral edges.

Prove that AD is congruent to A'D'.
Proof. Lettering the figure as shown above, we see that

AB is ll to A'B', BC is ll to B'C', CD is ll to c'D', and so on for all the corresponding sides.

$ 55 If two II planes are cut by a third plane, the lines

of intersection are II. Then AB=A'B', BC=B'C', CD=C'D', and so on for all the corresponding sides.

$ 10, 3 The opposite sides of a D are equal ..., Also, ZCBA=LC'B'A', ZDCB=ZD'C'B', and so on for all the corresponding {s.

$ 60 . . AD is congruent to A'D'.

$ 7,1 Since all their corresponding parts are equal, the sections can be

made to coincide by superposition. As a special case of this theorem, all right sections of a prism are congruent.

Proposition 2. Lateral Area

93. Theorem. The lateral area of a prism is the product of a lateral edge and the perimeter of a right section.

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Given VY, a rt. section of the prism AD'; L, the lateral area; e, a lateral edge; and p, the perimeter of the rt. section. Prove that

L= ep.
Proof. Lettering the figure as shown, we see that
AA'=BB'=CC'=DD'=EE'=e.

$ 86 Also, VW is I to BB'.

$ 33 Similarly, WX is I to CC', and so on.

... OAB'=BB':VW= e.VW. $ 25, 2 Similarly, OBC'=CC'.WX=eWX, and so on for all the lateral faces. Now L is the sum of these areas.

$ 86 L= e(VW+WX+ XY+YZ+ZV). Ax. 1 But VW+WX+ XY +YZ + ZV=p. ..L=ep.

Ax. 5

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