ELEMENTS OF PLANE GEOMETRY. BOOK I. DEFINITIONS. 1. Geometry is the science which treats of the properties and relations of magnitudes of space. Space has extension in all directions; but for the purpose of determining the size of portions of space, we consider it as having three dimensions, namely, length, breadth, and thickness. 2. A Point is position without size. 3. A Line is that which has but one dimension, namely, length. A line may be conceived as traced by a moving point. Lines are straight or curved. 4. A Straight Line is one which has the same direction at all its points. 5. A Curved Line is one which changes its direction at all its points. When the sense is ob vious, the word line, alone, is used for straight line, and the word curve, alone, for curved line. 6. A Surface is that which has only two dimensions, length and breadth. A surface may be conceived as generated by a moving line. Surfaces are plane or curved. 7. A Plane Surface, or a Plane, is a surface with which a straight line can be made to coincide in any direction. 8. A Curved Surface is a surface no portion of which is a plane. 9. A Solid is that which has three dimensions, length, breadth, and thickness. A solid may be conceived as generated by a moving surface. Points, lines, surfaces, and solids are the concepts of Geometry, and may be said to constitute the subject-matter of the science. 10. A Figure is some definite form of magnitude. 11. Lines, surfaces, and solids are called figures when reference is had to their form. 12. A Plane Figure is one, all of whose points are in the same plane. 13. PLANE GEOMETRY treats of plane figures. 14. Equal Figures are such as have the same form and size, that is, such as fill exactly the same space. 15. Equivalent Figures are such as have equal magnitudes. 16. Similar Figures are such as have the same form, although they may have different magnitudes. 17. A Plane Angle, or an Angle, is the opening between two lines which meet each other. The point in which the lines meet is called the Vertex, and the lines are called the sides of the angle. A plane angle is a species of surface. An angle is designated by placing a letter at each end of its sides, and one at its vertex, or by placing a small letter in it near the vertex. The latter is the method employed in this book, whenever it is convenient. In reading, when there is but one angle, we may name the letter at the vertex; but when there are two or more vertices at the same point, we name the three letters, with the one at the vertex between the other two: we may, however, in either case, simply name the letter placed in it. Thus, in Fig. 1, we say angle C, or angle a. In Fig. 2, G being the common vertex, we must say angle DGF, or angle b. The size of an angle depends upon the extent of opening of its sides, and not upon the length of the sides. 18. Adjacent Angles are such as have a common vertex and one common side between them. Thus, the angles a and b are adjacent angles. 19. A Right Angle is an angle included between two straight lines which meet each other so as to make the adjacent angles equal. Thus, if the angles a and b are equal, each is a right angle. 20. Perpendicular Lines are such as make right angles with each other. 21. An Acute Angle is one which is less than a right angle; as angle a. 22. An Obtuse Angle is one which is greater than a right angle; as angle ABC. Acute and obtuse angles are called oblique angles. A B ab ab a C 23. Oblique Lines are lines which are not perpendicular to each other, and which meet if sufficiently produced. 24. Two angles are Complements of each other when their sum is equal to a right angle. Thus, angle a is the complement of angle b, and angle b is the complement of angle a. 25. Two angles are Supple- A ments of each other when their sum is equal to two right angles. Thus, angle a is the supplement of angle ABC, and angle ABC is the supplement of angle a. a 27. If two lines are cut by a third line, eight angles are formed, which are named as follows: Angles a, b, c, and d are Exterior ab Angles. Angles e, f, g, and h are e/f Interior Angles. The pairs of an g/h c/d gles a and d, b and, are Alternate Exterior Angles. The pairs of an gles e and h, f and g, are Alternate Interior Angles. The pairs of angles a and g, band h, e and c, f and d, are cor responding Angles. 28. Parallel Straight Lines are such as lie in the same plane and cannot meet how far soever they are produced either way. They have the same direction. 29. A Circle is a plane figure bounded by a curve, all the points of which are equally distant from a point within, called the Centre. The Circumference of a circle is the curve which bounds it. A Radius of a circle is a line extending from the centre to any point in the circumference. The diagram represents a circle whose B D centre is O. The curve ABCD is the circumference, and the line OA is a radius. |