Proposition XXIV 498. Represent a trihedral angle. How does the sum of any two of its face angles compare with the third face angle? Theorem. The sum of any two face angles of a trihedral angle is greater than the third face angle. Data: Any trihedral angle, as Q-ABC, having one face angle, as AQC, greater than either of the other face angles. To prove ZAQB + ZBQC greater than ▲ AQC. D Proof. In the face AQC draw QD, making ZAQD: = ZAQB; through any point, as D, of QD draw ADC in the plane AQC; take QB = QD, and through line AC and point B pass a plane. .. Ax. 4, ZAQB + BQC is greater than ≤ AQD + ≤ DQC, ZAQB + BQC is greater than ▲ AQC. or Therefore, etc. ZAQB = ZAQD; Why? Why ? Q.E.D. Proposition XXV 499. Represent any convex polyhedral angle; around some point in a plane as a common vertex construct in succession angles equal to the face angles of this polyhedral. How does their sum compare with four right angles? Theorem. The sum of the face angles of any convex polyhedral angle is less than four right angles. Datum: Any convex polyhedral angle, as Q. To prove that the sum of the face angles of Q is less than four right angles. Proof. Pass a plane intersecting the edges of Q in A, B, C, etc. From 0, any point within ABCDE, draw OA, OB, OC, etc. The number of triangles whose common vertex is Q equals the number whose common vertex is 0. Hence, the sum of the angles of the triangles whose vertex is Q equals the sum of the angles of the triangles whose vertex is 0. But in the trihedral angles whose vertices are A, B, C, etc., § 498, ≤ QBA + ≤ QBC is greater than ▲ ABC, or ≤ ABO + ≤ CBO, and ≤ QCB + ≤ QCD is greater than ▲ BCD, or ▲ BCO + ≤ DCO. Hence, reasoning in a similar manner regarding the other base angles of the triangles, the sum of the base angles of all the triangles whose vertex is Q is greater than the sum of the base angles of the triangles whose vertex is 0. Therefore, the sum of the face angles at Q is less than the sum of the angles at 0. Why? But the sum of the angles at O equals four right angles. Hence, the sum of the face angles of Q is less than four right angles. Therefore, etc. Proposition XXVI Q.E.D. 500. Represent two trihedral angles having the three face angles of one equal respectively to the three face angles of the other. How do the trihedrals compare? Can there be two trihedrals which fulfill the same conditions and yet not be equal? What name is given to such trihedrals? Theorem. Two trihedral angles are either equal or symmetrical, if the three face angles of one are equal to the three face angles of the other, each to each. Data: Any two trihedral angles, as Q and Q', having the face angles AQB, BQC, AQC equal to the face angles A'Q'B', B'Q'C', A'Q'C', each to each. To prove Q either equal or symmetrical to Q'. Proof. On the edges of Q and Q' take the equal distances Q4A, QB, QC, Q'A', Q'B', Q'C', and draw AB, BC, AC, A'B', B'C', A'C'. Then, ▲ QAB, QBC, QAC are equal to ▲ Q'A'B', Q'B'C', Q'A'C', each to each. Hence, ▲ ABC = ▲ A'B'C', and ▲ BAC = ▲ B'A'C'. Why? Why? On the edge QA take AD and on Q'A' take A'D' =AD. At D and D' construct the plane HDK and H'D'K' of the dihedrals Q4 and Q'A' respectively, the sides meeting AB, AC', A'B', and A'C' as at H, K, H' and K' respectively, inasmuch as QAB, QAC, etc., are acute, being angles of isosceles A QAB, etc. Draw HK and H'K'. but L BAC B'A'C'; = ▲ AHK = ▲ A'H'K', and HK = H'K' ; DH= D'H', and DK = D'K' ; ▲ HDK= AH'D'K', and ▲ HDK = ZH'D'K'. Hence, § 477, dihedral QA = dihedral Q'A'. In like manner it may be shown that the dihedral angles QB and QC are equal to the dihedral angles Q'B' and Q'c' respectively. Hence, § 494, if the equal angles are arranged in the same order, as in the first two figures, the two trihedral angles are equal; but if they are arranged in the reverse order, as in the first and third figures, the two trihedral angles are symmetrical. Therefore, etc. Q.E.D. 501. 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. SUPPLEMENTARY EXERCISES Ex. 767. If a straight line is parallel to a plane, any plane perpendicular to the line is perpendicular to the plane. Ex. 768. If a straight line intersects two parallel planes it makes equal angles with them. Ex. 769. If a line is parallel to each of two planes, the intersections which any plane passing through it makes with the planes are parallel. Ex. 770. The projections of parallel straight lines on any plane are either parallel or coincident. Ex. 771. Find the locus of points which are equidistant from three given points not in the same straight line. Ex. 772. From any point within the dihedral angle A-BC-D, EF and EG are drawn perpendicular to the faces AC and BD, respectively, and GH perpendicular to AC at H. Prove that FH is perpendicular to BC. Ex. 773. If a plane is passed through the middle point of the common perpendicular to two straight lines in space, and parallel to both lines, it bisects every straight line drawn from any point in one line to any point in the other line. Ex. 774. If the intersections of several planes are parallel, the perpendiculars drawn to them from any point lie in one plane. Ex. 775. If two face angles of a trihedral are equal, the dihedral angles opposite them are also equal. Ex. 776. A trihedral angle, having two of its dihedral angles equal, may be made to coincide with its symmetrical trihedral angle. Ex. 777. In any trihedral the three planes bisecting the three dihedrals intersect in the same straight line. Ex. 778. In any trihedral the planes which bisect the three face angles, and are perpendicular to those faces, respectively, intersect in the same straight line. BOOK VIII POLYHEDRONS 502. A solid bounded by planes is called a Polyhedron. The intersections of the planes which bound a polyhedron are called its edges; the intersections of the edges are called its vertices; and the portions of the planes included by its edges are called its faces. The line joining any two vertices of a polyhedron, not in the same face, is called a diagonal of the polyhedron. 503. A polyhedron having four faces is called a tetrahedron; one having six faces is called a hexahedron; one having eight faces is called an octahedron; one having twelve faces is called a dodecahedron; one having twenty faces is called an icosahedron. 504. If the section made by any plane cutting a polyhedron is a convex polygon, the solid is called a Convex Polyhedron. Only convex polyhedrons are considered in this work. PRISMS 505. A polyhedron two of whose faces are equal polygons, which lie in parallel planes and have their homologous sides parallel, and whose other faces are parallelograms, is called a Prism. The two equal and parallel faces of the prism are called its bases; the other faces are called lat eral faces; the intersections of the lateral faces are called lateral edges; the sum of the lateral faces is called the lateral, or convex surface; and the sum of the areas of the lateral faces is called the lateral area of the prism. The lateral edges of a prism are parallel and equal. § 153. The perpendicular distance between the bases of a prism is its altitude. |