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 = 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 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.

E

CD B

A

F

FIG. 8.

a

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 A and AOD (Fig. 10).

D

B

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).

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. 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.

D

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.

75. 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; as, angles a and b, c and d (Fig. 16).

76. Two angles are called complementary when their sum is equal to a right angle; and each is called the complement of the other; as angles DOB and DOC (Fig. 17).

77. 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. 18).

UNIT OF ANGLES.

FIG. 16.

B FIG. 17.

B

FIG. 18.

78. By adopting a suitable unit for measuring angles we are able to express the magnitudes of angles by numbers. If we suppose OC (Fig. 15) to turn about O from coincidence with OA until it makes one three hundred sixtieth of a revolution, it generates an angle at O, which is taken as the unit for measuring angles. This unit is called a degree.

The degree is subdivided into sixty equal parts, called minutes, and the minute into sixty equal parts, called seconds. Degrees, minutes, and seconds are denoted by symbols. Thus, 5 degrees 13 minutes 12 seconds is written 5° 13' 12".

A right angle is generated when OC has made one fourth of a revolution and contains 90°; a straight angle, when OC has made half of a revolution and contains 180°; and a perigon, when OC has made a complete revolution and contains 360°. NOTE. The natural angular unit is one complete revolution. this unit would require us to express the values of most angles by fractions. The advantage of using the degree as the unit consists in its convenient size, and in the fact that 360 is divisible by so many different integral numbers.

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