THE STRAIGHT LINE. 44. Postulate. A straight line can be drawn from one point to another. 45. Postulate. A straight line can be produced indefinitely. 46. Axiom.* Only one straight line can be drawn from one point to another. Hence, two points determine a straight line. 47. COR. 1. Two straight lines which have two points in common coincide and form but one line. 48. COR. 2. Two straight lines can intersect in only one point. For if they had two points common, they would coincide and not intersect. Hence, two intersecting lines determine a point. 49. Axiom. A straight line is the shortest line that can be drawn from one point to another. 50. DEF. The distance between two points is the length of the straight line that joins them. 51. A straight line determined by two points may be considered as prolonged indefinitely. 52. If only the part of the line between two fixed points is considered, this part is called a segment of the line. 53. For brevity, we say "the line AB," to designate a segment of a line limited by the points A and B. 54. If a line is considered as extending from a fixed point, this point is called the origin of the line. * The general axioms on page 6 apply to all magnitudes. Special geometrical axioms will be given when required. 55. If any point, C, is taken in a given straight line, AB, the two parts CA and CB are said to have opposite directions from the point C (Fig. 5). A← B FIG. 5. Every straight line, as AB, may be considered as extending in either of two opposite directions, namely, from A towards B, which is expressed by AB, and read segment AB; and from B towards A, which is expressed by BA, and read segment BA. 56. If the magnitude of a given line is changed, it becomes longer or shorter. Thus (Fig. 5), by prolonging AC to B we add CB to AC, and AB = ACCB. By diminishing AB to C, we subtract CB from AB, and ACAB CB. If a given line increases so that it is prolonged by its own magnitude several times in succession, the line is multiplied, and the resulting line is called a multiple of the given line. Lines of given length may be added and subtracted; they may also be multiplied by a number. THE PLANE ANGLE. 57. The opening between two straight lines drawn from the same point is called a plane angle. The two lines, ED and EF, are called the sides, and E, F the point of meeting, is called the vertex of the angle. FIG. 7. The size of an angle depends upon the extent of opening of its sides, and not upon the length of its sides. 58. If there is but one angle at a given vertex, the angle is designated by a capital letter placed at the vertex, and is read by simply naming the letter. If two or more angles have the same vertex, each angle is designated by three letters, and is read by naming the three letters, the one at the vertex between the others. Thus, DAC (Fig. 8) is the angle formed by the sides AD and AC. An angle is often designated by placing a small italic letter between the sides and near the vertex, as in Fig. 9. CD B V A FIG. 8. α A B FIG. 9. 59. Postulate of Superposition. Any figure may be moved from one place to another without altering its size or shape. 60. The test of equality of two geometrical magnitudes is that they may be made to coincide throughout their whole extent. Thus, Two straight lines are equal, if they can be placed one upon the other so that the points at their extremities coincide. Two angles are equal, if they can be placed one upon the other so that their vertices coincide and their sides coincide, each with each. 61. A line or plane that divides a geometric magnitude into two equal parts is called the bisector of the magnitude. If the angles BAD and CAD (Fig. 8) are equal, AD bisects the angle BAC. 62. Two angles are called adjacent angles when they have the same vertex and a common side between them; as the angles BOD and AOD (Fig. 10). 63. When one straight line meets another straight line and makes the adjacent angles equal, each of these angles is called a right angle; as angles DCA and DCB (Fig. 11). 64. A perpendicular to a straight line is a straight line that makes a right angle with it. Thus, if the angle DCA (Fig. 11) is a right angle, DC is perpendicular to AB, and AB is perpendicular to DC. 65. The point (as C, Fig. 11) where a perpendicular meets another line is called the foot of the perpendicular. 66. When the sides of an angle extend in opposite directions, so as to be in the same straight line, the angle is called a straight angle. Thus, the angle formed at C (Fig. 12) with its sides CA and CB extending in opposite directions from C is a straight angle. 67. COR. A right angle is half a straight angle. 68. An angle less than a right angle is called an acute angle; as, angle A (Fig. 13). 69. An angle greater than a right angle and D less than a straight angle is called an obtuse angle; as, angle AOD (Fig. 14). FIG. 13. FIG. 14. 70. An angle greater than a straight angle and less than two straight angles is called a reflex angle; as, angle DOA, indicated by the dotted line (Fig. 14). 71. Angles that are neither right nor straight angles are called oblique angles; and intersecting lines that are not perpendicular to each other are called oblique lines. EXTENSION OF THE MEANING OF ANGLES. 72. Suppose the straight line OC (Fig. 15) to move in the plane of the paper from coincidence with OA, about the point O as a pivot, to the position OC; then the line OC describes or generates the angle AOC, and the magnitude of the angle AOC depends upon the amount of rotation of the line from the position OA to the position OC. B B FIG. 15. If the rotating line moves from the position OA to the position OB, perpendicular to OA, it generates the right angle AOB; if it moves to the position OD, it generates the obtuse angle AOD; if it moves to the position OA', it generates the straight angle AOA'; if it moves to the position OB', it generates the reflex angle AOB', indicated by the dotted line; and if it moves to the position OA again, it generates two straight angles. Hence, 73. The angular magnitude about a point in a plane is equal to two straight angles, or four right angles; and the angular magnitude about a point on one side of a straight line drawn through the point is equal to a straight angle, or two right angles. 74. The whole angular magnitude about a point in a plane is called a perigon; and two angles whose sum is a perigon are called conjugate angles. NOTE. This extension of the meaning of angles is necessary in the applications of Geometry, as in Trigonometry, Mechanics, etc. |