PLANE GEOMETRY INTRODUCTION DEFINITIONS 1. A physical body, such as a block of wood or iron, occupies a definite portion of space. The portion of space occu pied by a physical body is called a geometric solid or a solid. D F E A E of space. It has 2. DEF. A solid is a limited portion three dimensions, length, breadth, and thickness. 3. DEF. Surfaces are the boundaries of solids; as ABED or BEFC. They have two dimensions, length and breadth. The boundary between a window pane and the air is a surface. Obviously such a boundary has no thickness. 4. DEF. Lines are the boundaries of surfaces, as AB, AD. (Figure of § 1.) Lines have but one dimension, length. Thus, the annexed black line AB is not a geometric line, for it has breadth. A B A true geometric line, however, is represented by the boundary between the black and the white. 5. DEF. Points are the boundaries or the extremities of lines. They are without dimensions, having position only. Surfaces may be conceived as existing independent of the solids whose boundaries they form. In like manner, lines and points may exist independently in space. 6. DEF. A geometric figure is a point, line, surface, or solid, or any combination of any or all of these; as M or N. A rectilinear figure is a figure composed of straight lines only. 7. DEF. Geometry is the science that treats of the properties of geometric figures. 8. The simplest line is a straight line. It is represented approximately by a string stretched taut between two points; as AB. The word "line" is frequently used to denote a straight line. A B لله A very simple term is, as a rule, not easy to define on account of the difficulty of finding still simpler terms by which to define it. The notion of a straight line is such a simple and fundamental one that it is practically impossible to give a good definition of it. 9. DEF. A curved line or curve is a line no portion of which is straight; as CD. 10. DEF. A broken line is a line composed of different successive straight lines; as EF. 11. The expression, straight line, is used to denote both an unlimited straight line and a part of such line. A line of definite length, also called a segment, or line-segment, is represented by a line whose ends are marked; as AB. The length of this line is also called the distance from A to B. A B A line whose ends are not marked represents a line of indefinite length; as CD. 12. The direction of the line AB means the direction from A toward B; of BA, the direction from B toward A. 13. To produce the line AB means to prolong it through B; to produce BA means to prolong it through 4. 14. DEF. A plane surface or a plane is a surface such that a straight line joining any two of its points lies entirely in the surface. 15. DEF. A plane figure is a geometric figure, all of whose points lie in the same plane; as EF. E 16. DEF. Plane Geometry treats of plane figures only. 17. DEF. Solid Geometry treats of figures which are not plane. 18. When one figure can be placed upon another so that each point of one lies upon some point of the other, the figures are said to coincide. 19. DEF. Congruent figures are those that can be made to coincide. For reasons that will appear later congruent lines are frequently called equal lines. Similarly angles that can be superposed are usually called equal angles. (See note, p. 205.) 20. Proof by superposition is the method of proving the congruence of two figures by making them coincide. 21. To bisect a line means to divide it into two equal parts. Thus, AC is bisected if AD = DC. A D |