553. DEF. Two polyhedral angles are vertical if the edges of each are prolongations of the edges of the other.

PROPOSITION XXVIII

554. Two trihedral angles are symmetric:

(1) If two face angles and the included dihedral angle of the one are respectively equal to two face angles and the included dihedral angle of the other, or, (2) If two dihedral angles and the included face angle of the one are respectively equal to two dihedral angles and the included face angle of the other,

or, (3) Three face angles of the one are equal respectively to three face angles of the other,

provided all equal parts are arranged in reverse order.

444

Proof.

C

If A and B are the given trihedral angles, construct

trihedral angle C symmetric to 4.

Since C and 4 have all parts arranged in reverse order, C and B have all parts arranged in the same order.

But since C is symmetric to 4, the congruent figure B is symmetric to 4.

555. COR. Vertical trihedral angles are symmetric.

Q. E. D.

556. NOTE. There is a remarkable analogy between triangles and trihedral angles (also between polygons and polyhedral angles). The sides of a triangle correspond to the face angles, and the angles of a triangle correspond to the dihedral angles of a trihedral angle. Proofs of theorems that relate to trihedral angles may often be found from the proofs of the analogous triangle proposition by the addition of the letter V (i.e. the vertex) to each symbol. Thus, the following statement, if we erase every where the letter V, is the proof of the proposition :

If two sides (AB and BC) of a triangle (ABC) are equal, the opposite angles (A and B) are equal. If we include the letter V, it is the proof for the theorem :

If two face angles of a trihedral angle are equal, the opposite dihedral angles are equal.

.. V-ABD is symmetric (respectively congruent)

Ex. 1479. State proposition relating to trihedral or polyhedral angles that correspond to the following theorems of plane geometry :

(a) The sum of two sides of a triangle is greater than the third side. (b) If two angles of a triangle are equal, the opposite sides are equal. (c) If two sides of a triangle are unequal, the opposite angles are unequal, etc.

(d) Two triangles are congruent if two sides and the included angle of the one, etc.

(e) If the opposite sides of a quadrilateral are equal, the opposite angles are equal.

Ex. 1480. In the diagram for Prop. XXIII,

ZAVD + 2DVB <▲ AVC + ▲ CVB.

Ex. 1481. If in tetrahedral angle V-ABCD, ▲ AVB = LAVD and ZBVC = 2CVD, then dihedral angle VD = dihedral angle VB.

Ex. 1482. If the opposite face angles of a tetrahedral angle are equal, the opposite dihedral angles are equal.

Ex. 1483. The projections of parallel lines upon the same plane are parallel.

Ex. 1484. Planes bisecting supplementary adjacent dihedral angles are perpendicular to each other.

Ex. 1485. A line is twice as long as its projection upon a plane. What is the inclination of the line to the plane?

Ex. 1486. If the three face angles of a trihedral angle are equal, the three dihedral angles are equal.

Ex. 1487. If the three face angles of a trihedral angle are right angles, the three dihedral angles are right ones (a trirectangular trihedral angle).

Ex. 1488. If a straight line intersects two parallel planes, it makes equal angles with both planes.

Ex. 1489. Vertical polyhedral angles are symmetric.

Ex. 1490. Find the locus of a point equidistant from the three faces of a trihedral angle.

Ex. 1491. Find a point equidistant from four given points, not in one plane.

Ex. 1492. What is the locus of a point X, if Xlies in plane MN, PX equals a given line, and P is a point without MN?

Ex. 1493. If the sum of the four angles of a quadrilateral is equal to four right angles, its vertices lie in a plane.

Ex. 1494. If D is a point within a trihedral angle V-ABC, then ▲ AVD + 2 BVD+▲ CVD > { (< AVB + 2 BVC+2 C VA).

BOOK VII

POLYHEDRONS, CYLINDERS, AND CONES

POLYHEDRONS

557. DEF. A polyhedron is a solid bounded by planes. The faces of a polyhedron are the bounding planes; the edges are the intersections of the faces; and the vertices are the intersections of the edges.

558. DEF. A tetrahedron is a polyhedron of four faces; a hexahedron, one of six faces; an octahedron, one of eight faces; a dodecahedron, one of twelve faces; an icosahedron, one of twenty faces.

NOTE. The least number of faces that a polyhedron can have is four; for three planes intersecting in a common point form a trihedral angle, and, therefore, one more plane is needed to form a solid.

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

560. DEF. A convex polyhedron is one, every section of which is a convex polygon.

NOTE. All the polyhedrons treated of in this book are convex.

PRISMS AND PARALLELOPIPEDS

561. DEF. A prismatic surface is a surface generated by a moving straight line which continually intersects the boundary of a given polygon (as ABCDE) and is parallel

to a fixed straight line not in the plane of the polygon.

D

562. DEF. A prism is a polyhedron bounded by a prismatic surface and two parallel planes. A The faces made by the parallel planes are the bases; the faces made by the prismatic surface are the lateral faces. The lateral edges are the intersections of two lateral faces. The lateral area is the sum of the areas of the lateral faces. The altitude of a prism is the perpendicular distance between the planes of the bases. Obviously all lateral faces are parallelograms (483) and the lateral edges are equal and parallel.

563. DEF. A right prism is a prism whose lateral edges are perpendicular to the planes of the bases.

In a right prism all lateral faces are rectangles which are perpendicular to the base, and all lateral edges are equal to the altitude.

564. DEF. A regular prism is a right prism whose bases are regular polygons.

565. DEF. An oblique prism is a prism whose lateral edges are not perpendicular to the planes of the bases.

A prism is triangular, quadrangular, etc., according as its bases are triangles, quadrilaterals, etc.