34. A straight line determined by two points is considered as prolonged indefinitely both ways. Such a line is called an indefinite straight line. 35. Often only the part of the line between two fixed points is considered. This part is then called a segment of the line. For brevity, we say "the line AB" to designate a segment of a line limited by the points A and B. 36. Sometimes, also, a line is considered as proceeding from a fixed point and extending in only one direction. This fixed point is then called the origin of the line. 37. If any point C be taken in a given straight line AB, the two parts CA and CB are said to have opposite directions from the point C. A C FIG. 5. -B 38. Every straight line, as AB, may be considered as having opposite directions, namely, from A towards B, which is expressed by saying "line AB"; and from B towards A, which is expressed by saying "line BA." 39. 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 AC+ CB. By diminishing AB to C, we subtract CB from AB, and AC AB - CB. = If a given line increases so that it is prolonged by its own magnitude several times in succession, the line is multi A B C Ꭰ + E plied, and the resulting line is called a multiple of the given line. AB BC= CD DE, then AC-2AB, Hence, FIG. 6. Thus (Fig. 6), if AD=3 AB, and AC, AB AD, and AB={AE. Lines of given length may be added and subtracted; they may also be multiplied and divided by a number. PLANE ANGLES. 40. A plane angle is the difference in direction of two lines. The two lines are called the sides of the angle, and the point where the sides meet is called the vertex of the angle. 41. If there is but one angle at a given vertex, it is designated by a capital letter placed at the vertex, and is Aread by simply naming the letter; as, angle A (Fig. 7). But when two or more angles have the same vertex, each angle is designated by three letters, as shown in Fig. 8, and is read by naming the three letters, the one at the vertex be- A tween the others. Thus, the angle DAC means the angle formed by the sides AD and AC. It is often convenient to designate an angle by placing a small italic letter between the sides and near the vertex, as in Fig. 9. 42. Two angles are equal if they can be made to coincide. FIG. 7. FIG. 8. d α FIG 9. -D B 43. If the line AD (Fig. 8) is drawn so as to divide the angle BAC into two equal parts, BAD and CAD, AD is called the bisector of the angle BAC. In general, a line that divides a geometrical magnitude into two equal parts is called a bisector of it. C 46. 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. 11) with its sides CA and CB extending in opposite directions from C, is a straight angle. Hence a right angle may be defined as half a straight angle. 47. 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. 48. The point (as C, Fig. 11) where a perpendicular meets another line is called the foot of the perpendicular. 49. Every angle less than a right angle is called an acute angle; as, angle A. A FIG. 12. 50. Every angle greater than a right angle and less than a straight angle is called an obtuse angle; as, angle C (Fig. 13). 51. Every angle greater than a straight angle and less than two straight angles is called a reflex angle; as, angle O (Fig. 14). B D A FIG. 13. FIG. 14. 52. Acute, obtuse, and reflex angles, in distinction from right and straight angles, are called oblique angles; and intersecting lines that are not perpendicular to each other are called oblique lines. A D 53. When two angles have the same vertex, and the sides of the one are prolongations of the sides of the other, they are called vertical angles. Thus, a and b (Fig. 15) are vertical angles. 54. Two angles are called complementary when their sum a b d C B FIG. 15. is equal to a right angle; and each is called the complement of the other; as, angles DOB and DOC (Fig. 10). 55. Two angles are called supplementary when their sum is equal to a straight angle; and each is called the supplement of the other; as, angles DOB and DOA (Fig. 10). MAGNITUDE OF ANGLES. 56. The size of an angle depends upon the extent of opening of its sides, and not upon their length. Suppose the straight line OC 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 B' FIG. 16. 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 continues its rotation to the position OA, whence it started, it generates two straight angles. Hence the whole 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 that point is equal to one straight angle, or two right angles. Angles are magnitudes that can be added and subtracted; they may also be multiplied and divided by a number. |