PROPOSITION XXVII. THEOREM. 581. The sum of the face angles of any convex polyhedral angle is less than four right angles. Let S be a convex polyhedral angle, and let all its edges be cut by a plane, making the section ABCDE. To prove ASB + Z BSC, etc., less than four rt. 4. Proof. From any point O within the polygon draw OA, OB, OC, OD, OE. The number of the A having the common vertex O is the same as the number having the common vertex S. Therefore, the sum of the of all the ▲ having the common vertex S is equal to the sum of the of all the ▲ having the common vertex 0. But in the trihedral formed at A, B, C, etc., SAESAB is greater than ▲ SBA + ≤ SBC is greater than EAB, ABC, etc. § 580 Hence, the sum of the at the bases of the A whose common vertex is S is greater than the sum of the at the bases of the whose common vertex is 0. Ax. 4 Therefore, the sum of the at the vertex S is less than the sum of the at the vertex 0. But the sum of the at O is equal to 4 rt. . Ax. 5 § 88 . Q. E. D. 582. Two trihedral angles are equal or symmetrical when the three face angles of the one are respectively equal to the three face angles of the other. AAA D'C B'E' A BE Α E B' In the trihedral angles S and S', let the angles ASB, ASC, BSC be equal to the angles A'S'B', A'S'C', B'S'C', respectively. To prove that S and S' are equal or symmetrical. Proof. On the edges of these angles take the six equal distances SA, SB, SC, S'A', S'B', S'C'. Draw AB, BC, AC, A'B', B'C', A'C'. The isosceles A SAB, SAC, SBC are equal, respectively, to the isosceles A S'A'B', S'A'C', S'B'C'. § 143 .. AB, BC, CA are equal, respectively, to A'B', B'C', C'A'. .. ΔΑΒΟ = =A A'B'C'. § 150 At any point D in SA draw DE in the face ASB and DF in the face ASC 1 to SA. These lines meet AB and AC, respectively, (since the SAB and SAC are acute, each being one of the equal ▲ of an isosceles A). Draw EF. On A'S' take A'D' equal to AD. Draw D'E' in the face A'S'B' and D'F" in the face A'S'C' L to = For ED E'D', DF = D'F', and EF = E'F'. (since & EDF and E'D'F', the measures of these dihedral 4, are equal). In like manner it may be proved that the dihedral angles A-BS-C and A-CS-B are equal, respectively, to the dihedral angles A'-B'S'-C' and A'-C'S'-B'. .. S and S' are equal or symmetrical. §§ 577, 579 Q. E. D. This demonstration applies to either of the two figures denoted by S'-A'B'C', which are symmetrical with respect to each other. If the first of these figures is taken, S and S' are equal. If the second is taken, S and S' are symmetrical. 583. COR. If two trihedral angles have three face angles of the one equal to three face angles of the other, then the dihedral angles of the one are respectively equal to the dihedral angles of the other. THEOREMS. Ex. 616. Find the locus of points in space equidistant from two given intersecting lines. Ex. 617. Find the locus of points in space equidistant from all points in the circumference of a circle. Ex. 618. Find the locus of points in a plane equidistant from a given point without the plane. Ex. 619. Find a point at equal distances from four points not all in the same plane. Ex. 620. Two dihedral angles which have their edges parallel and their faces perpendicular are equal or supplementary. Ex. 621. The projections on a plane of equal and parallel lines are equal and parallel. Ex. 622. If two face angles of a trihedral angle are equal, the dihedral angles opposite them are equal. Ex. 623. The planes that bisect the dihedral angles of a trihedral angle intersect in the same straight line. Ex. 624. If the face angle ASB of the trihedral angle S-ABC is bisected by the line SD, the angle CSD is less than half the sum of the angles ASC and BSC. Ex. 625. An isosceles trihedral angle and its symmetrical trihedral angle are superposable. Ex. 626. Find the locus of points equidistant from the three edges of a trihedral angle. Ex. 627. Find the locus of points equidistant from the three faces of a trihedral angle. Ex. 628. Two trihedral angles are equal when two dihedral angles and the included face angle of the one are equal, respectively, to two dihedral angles and the included face angle of the other and similarly placed. Ex. 629. Two trihedral angles are equal when two face angles and the included dihedral angle of the one are equal, respectively, to two face angles and the included dihedral angle of the other and similarly placed. BOOK VII. POLYHEDRONS, CYLINDERS, AND CONES. POLYHEDRONS. 584. DEF. A polyhedron is a solid bounded by planes. The bounding planes are called the faces, the intersections of the faces, the edges, and the intersections of the edges, the vertices, of the polyhedron. 585. DEF. A diagonal of a polyhedron is a straight line joining any two vertices not in the same face. 586. DEF. A section of a polyhedron is the figure formed. by its intersection with a plane passing through it. 587. DEF. A polyhedron is convex if every section is a convex polygon. Only convex polyhedrons are considered in this work. 588. DEF. A polyhedron of four faces is called a tetrahedron; one of six faces, a hexahedron; one of eight faces, an octahedron; one of twelve faces, a dodecahedron; one of twenty faces, an icosahedron. NOTE. Full lines in the figures of solids represent visible lines, dashed lines represent invisible lines. |