2d. If three points, A, B, C, not in the same straight line are given, any two of them, as A and B, may be joined by a straight line; then this is the same as the 1st case. 3d. If two intersecting lines A B, AC are given, any point, C, out of the line AB can be taken in the line A C; then the plane passing through the line AB and the point C contains the two lines A B and A C, and is determined by them. 5. Corollary. The intersection of two planes is a straight line; for the intersection cannot contain three points not in the same straight line, since only one plane can contain three such points. THEOREM II. 6. Oblique lines from a point to a plane equally distant from the perpendicular are equal; and of two oblique lines unequally distant from the perpendicular, the more remote is the greater. (I. 51); and A E=AD= AC; therefore AF> AD or AC. 7. Cor. 1. Conversely, equal oblique lines from a point to a plane are equally distant from the perpendicular; therefore they meet the plane in the circumference of a circle whose centre is the foot of the perpendicular. Of two unequal lines the greater is more remote from the perpendicular. 8. Cor. 2. The perpendicular is the shortest distance from a point to a plane. E THEOREM III. 9. The intersections of two parallel planes with a third plane are parallel. Let A B and CD be the intersec- M tions of the plane A D with the parallel planes MN and PQ; then A B and CD are parallel. B N D P For the lines A B and CD cannot meet though produced indefinitely, since the planes MN and PQ in which they are cannot meet; and they are in the same plane AD; therefore they are parallel. C 10. Corollary. Parallels intercepted between parallel planes are equal. For the opposite sides of the quadrilateral A D being parallel, the figure is a parallelogram; therefore ACBD. THEOREM IV. 11. If two angles not in the same plane have their sides parallel and similarly situated, the angles are equal and their planes parallel. Let A B C and DEF be two angles M in the planes MN and PQ, having their sides A B, BC respectively parallel to D E, E F, and similarly situated; then 1st. The angles A B C and D E F are P equal. For, taking EDBA, and B A C N D Q and B F are parallelograms, since A B and B C are respectively equal and parallel to DE and EF; therefore AD and CF, being each equal and parallel to BE, are equal and parallel to each other; and therefore A F is a par- M allelogram, and AC is equal to DF; therefore the two triangles ABC and DEF, being mutually equilateral, are mutually equiangular, and the angles ABC and DEF are equal. 2d. The planes of these angles are parallel. For, since two intersecting lines determine a plane, the plane of P the lines A B and BC must be parallel to the plane of the lines DE and EF, as A B and B C are respectively parallel to DE and E F. THEOREM V. 12. If two straight lines are cut by parallel planes, they are divided proportionally. Let A B and C D be cut by the parallel M planes MN, PQ, and RS, in the points N in G, the plane of the lines A B and A DR cuts the parallel planes PQ and RS in B S EG and BD; therefore E G and BD are parallel (9), and we have (II. 16) AE: EBAG:GD The plane of the lines A D and CD cuts the parallel planes MN and PQ in AC and GF; therefore AC is parallel to G F; and we have AG: GDCF: FD Hence we have (Pn. 11) AE: EBCF:FD EXERCISES. The following Theorems, depending for their demonstration upon those already demonstrated, are introduced as exercises for the pupil. In some of them references are made to the propositions upon which the demonstration-depends. They are not connected with the propositions in the following books, and can be omitted if thought best. 13. An infinite number of planes can pass through a given line. (4.) 14. There can be but one perpendicular from a point to a plane. 15. A line perpendicular to each of two lines at their point of intersection is perpendicular to the plane of these lines. (4.) (I. 76.) 16. Parallel lines are equally inclined to the same plane. 17. State the converse of (16). Is it true? 18. Lines parallel to a line in a given plane are parallel to the plane. 19. State the converse of (18). Is it true? 20. Parallel planes are equally inclined to the same straight line. 21. State the converse of (20). Is it true? 22. Parallel lines included between parallel planes are equal. BOOK V. POLYEDRONS. DEFINITIONS. 1. A Polyedron is a solid bounded by planes. The bounding planes are called faces; their intersections, edges; the intersections of the edges, vertices. 2. The Volume of a solid is the measure of its magnitude. It is expressed in units which represent the number of times it contains the cubical unit taken as a standard. 3. Equivalent Solids are those which are equal in volume. 4. Similar Solids are those whose homologous lines have a constant ratio. (Corollary.) It follows that similar solids are bounded by the same number of similar polygons similarly situated. PRISMS AND CYLINDERS. 5. A Prism is a polyedron two of whose faces are equal polygons having their homologous sides parallel. (Corollary.) The other faces are parallelograms. The equal parallel polygons are called bases; as A B and CD. 6. The Altitude of a prism is the perpendic ular distance between its bases; as E F. |