5. Equals multiplied by equals give equals. 8. A straight line is the shortest distance between two points. 22. A Postulate is an assumption that a problem is possible. 23. POSTULATES 1. It is assumed that a straight line can be drawn between any two points, or can be produced to any required length. 2. It is assumed that a geometric figure, or portion of a figure, can be moved without changing the relations of its parts to each other. 24. ORDER OF WRITTEN PROOF A proposition may be divided into five parts: 1. Theorem (or Problem); 2. Application; 3. Construction; 4. Demonstration; 5. Conclusion. These may be followed by Corollaries and Scholia. 25. The Theorem (or Problem is the general statement of the proposition to be considered. 26. The Application is the particular statement of the proposition as applied to the figure which is used in the process of proof. 27. The Construction is the drawing of auxiliary lines or figures used in the proof, or the placing of figures in any desired position. 28. The Demonstration is the proof proper, and consists of the logical reasoning employed to establish the truth of the theorem. 29. The Conclusion is a final statement of the theorem as proved, and in written demonstration is often omitted. 30. Plane Geometry treats of figures whose elements are all in the same plane. 31. Solid Geometry treats of figures whose elements are not all in the same plane. 32. Plane figures are sometimes obtained by passing a plane through a geometric solid, and such figures are section views of the solid. 33. SIGNS AND ABBREVIATIONS Il parallel. circle. = arc. + plus. minus. equal. s angles. I perpendicular. Ŝ arcs. etc. 34. The same letter or figure used by repetition to designate two or more angles indicates that the angles are equal, and usually that they are equal by hypothesis in the discussion of any theorem. Such angles may be designated thus: 2, 2', 2", etc. The same mark, or the same B) с number of small cross lines placed upon two lines in a figure indicates that the two lines are equal, usually by hypothesis, as A D AB= CD. D B 4X II In Fig. 2 ABC is, by means of the mark of equality, shown to be an isosceles A (for definition see p. 24). Also, the values given to the ís of the isosceles A show that the Z at B is four times either of the Is A and C, and of course twice their sum. DC is I AC and is met by A AB produced. The figures thus marked are graphic pictures of geometric relations, and from them deductions may be made. Observe the bisection of lines, and the consequent equality of parts of lines, indicated in the accompanying figures : X X с FIG. 2. The equality of angles is shown in such figures as the following: Note. — The student should early accustom himself to the use of these and other symbols of equality. For example, a double cross may indicate a common line or angle; that is, a line or angle common to two distinct geometric figures, as the angle A, in the triangles ABC and ADE, in section 255. PLANE GEOMETRY BOOK I DEFINITIONS THE POINT 35. The position of a point is determined by its direction and distance from a known point; or its direction from two known points, provided the three points are not in the same straight line. Thus, the position of A is known, if its direction and distance from B are known, or the position of A'is determined if its direction from both B and C is known; BA and CA representing the direction of the point A from B and C respectively, A lies at B/ с the intersection of these two lines. THE STRAIGHT LINE 36. A straight line is determined in position by any two of its points, or by one point and its direction. 37. The Origin of a line is the point from which the line is supposed to be drawn. 7 |