526. Two triedral angles are equal if the three face angles of the one are respectively equal to the three face angles of the other, and all are arranged in the same order.

Hyp. In triedral angles V-ABC and V'-A'B'C',

LAVBLA'V'B', LBVC=ZB'V'C',

LCVA=LC'V'A'.

and

To prove

V-ABC-V'—A'B'C'.

Proof. On the six edges lay off

VA=VB=VC = V'A' =V'B' =V'C'.

Draw AB, BC, CA, A'B', B'C', and C'A'.

On AV and A'V', respectively, lay off AD equal to A'D'. Draw DE in face AVB, and DF in face AVC, perpendicular to VA. These lines meet AB and AC, respectively, since triangles AVB and AVC are isosceles.

In like manner, draw D'E' and D'F', and join EF and E'F'. [To be completed by the student.]

527. DEF. Vertical polyedral angles are polyedral angles which have a common vertex, and the edges of one are prolongations of the edges of the other.

PROPOSITION XXIX

528. Two triedral angles are symmetrical:

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

or, (2) If two diedral angles and the included face angle of the one are respectively equal to two diedral 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.

44

Proof. If A and B are the given triedral angles, construct triedral angle C symmetrical to A.

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

But since C is symmetrical to A, its equal B is symmetrical to A.

529. COR. Vertical triedral angles are symmetrical.

Ex. 1066. Vertical polyedral angles are symmetrical.

EXERCISES

Q.E.D.

Ex. 1067. If two face angles of a triedral angle are equal, the opposite diedral angles are equal.

Ex. 1068. Find the locus of a point, having a given distance from a given plane.

Ex. 1069. Planes bisecting supplementary adjacent diedral angles are perpendicular to each other.

Ex. 1070. The projections of parallel lines upon the same plane are parallel.

Ex. 1071. The bisectors of the three diedral angles of a triedral angle meet in the same straight line.

Ex. 1072. A line is twice as long as its projection upon a plane. What is the inclination of the line to the plane?

Ex. 1073. If the three face angles of a triedral angle are equal, the three diedral angles are equal.

Ex. 1074. If the opposite face angles of a tetraedral angle are equal, the opposite diedral angles are equal.

Ex. 1075. If the three face angles of a triedral angle are right angles, the three diedral angles are right ones (a tri-rectangular triedral angle). Ex. 1076. The bisecting planes of alternate interior diedral angles made by parallel planes are parallel to each other.

Ex. 1077. If a straight line intersects two parallel planes, it makes equal angles with both planes.

Ex. 1078. In a triedral angle, the greater face angle is opposite the greater diedral angle.

Ex. 1079. If the sum of the four angles of a quadrilateral is equal to four right angles, its vertices lie in a plane.

Ex. 1080. The common perpendicular is the shortest distance that can be drawn between two straight lines not in the same plane.

BOOK VII

POLYEDRONS, CYLINDERS, AND CONES

POLYEDRONS

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

531. DEF. A tetraedron is a polyedron of four faces; a hexaedron, one of six faces; an octaedron, one of eight faces; a dodecaedron, one of twelve faces; an icosaedron, one of twenty faces.

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

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

533. DEF. A convex polyedron is one, every section of which is a convex polygon.

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

PRISMS AND PARALLELOPIPEDS

534. DEF. A prism is a polyedron, two of whose faces are equal polygons in parallel planes, and whose other faces are parallelograms. The bases of a prism are the equal polygons; the lateral faces are the parallelograms; the lateral edges are the intersections of the lateral faces; and the lateral area is the sum of the areas of the lateral faces.

From definition we have the lateral edges of a prism are equal and parallel.

535. DEF. The altitude of a prism is the perpendicular distance between the planes of the bases.

536. DEF. A right prism is a prism whose lateral edges are perpendicular to the planes of the bases. In a right prism each lateral edge is equal to the altitude.

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

538. 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.

539. DEF. A truncated prism is the part of a prism included between a base and a section made by a plane not parallel to the base and cutting all of the lateral edges.