PLANE GEOMETRY INTRODUCTION 1. Geometry is a science which treats of the measurement of magnitudes. 2. A definition is a statement explaining the significance of a word or a phrase. Every definition should be clear, simple, descriptive, and correct; that is, it should contain the essential qualities or exclude all others, or both. 3. A point is that which has position but not magnitude. 4. A line is that which has length but no other magnitude. 5. A straight line is a line which is determined (fixed in position) by any two of its points. That is, two lines that coincide entirely, if they coincide at any two points, are straight lines. 6. A rectilinear figure is a figure containing straight lines and no others. 7. A surface is that which has length and breadth but no other magnitude. 8. A plane is a surface in which if any two points are taken, the straight line connecting them lies wholly in that surface. 9. Plane Geometry is a science which treats of the properties of magnitudes in a plane. 10. A solid is that which has length, breadth, and thickA solid is that which occupies space. ness. 11. Boundaries. The boundaries (or boundary) of a solid are surfaces. The boundaries (or boundary) of a surface are lines. The boundaries of a line are points. These boundaries can be no part of the things they limit. face is no part of a solid; a line is no part of a surface; a. point is no part of a line. A sur 12. Motion. If a point moves, its path is a line. Hence, if a point moves, it generates (describes or traces) a line; if a line moves (except upon itself), it generates a surface; if a surface moves (except upon itself), it generates a solid. NOTE. Unless otherwise specified the word "line" hereafter means straight line. 13. A plane angle is the amount of divergence of two straight lines that meet. The lines are called the sides of the angle. The vertex of an angle is the point at which the lines meet. 14. Adjacent angles are two angles that have the same. vertex and a common side between them. 15. Vertical angles are two angles that have the same vertex, the sides of one being prolongations of the sides of the other. 16. If one straight line meets another and makes the adjacent angles equal, the angles are right angles. 17. One line is perpendicular to another if they meet at right angles. Either line is perpendicular to the other. The point at which the lines meet is the foot of the perpendicular. Oblique lines are lines that meet but are not perpendicular. 18. A straight angle is an angle whose sides lie in the same straight line, but extend in opposite directions from the vertex. LV OBTUSE ANGLE ACUTE COMPLEMENTARY SUPPLEMENTARY ANGLES 19. An obtuse angle is an angle that is greater than a right angle. An acute angle is an angle that is less than a right angle. An oblique angle is any angle that is not a right angle. 20. Two angles are complementary if their sum is equal to one right angle. Two angles are supplementary if their sum is equal to two right angles. Thus the complement of an angle is the difference between one right angle and the given angle. The supplement of an angle is the difference between two right angles and the given angle. 21. A degree is one ninetieth of a right angle. The degree is the familiar unit used in measuring angles. It is evident that there are 90° in a right angle; 180° in two right angles, or a straight angle; 360° in four right angles. 22. Notation. A point is usually denoted by a capital letter, placed near it. A line is denoted by two capital letters, placed one at each end, or one at each of two of its points. Its length is sometimes represented advantageously by a small letter written near it. Thus, the line AB; the line RS; the line m. A R S B m |