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The point is called the Vertex, and the lines are called the Arms, of the angle. A line drawn from the vertex, and turning about the vertex in the plane of the angle from the position of coincidence with one arm to that of coincidence with the other, is said to turn through the angle; and the angle is greater as the quantity of turning is greater.
59. Since the line can turn from the one position to the other in either of two ways, two angles are formed by two lines drawn from a point.
Each of these angles is called the Explement of the other. If we say two lines going out from a point form an angle, we
are fixing the attention upon one of the two explemental angles which they really form; and usually we mean the smaller angle.
60. Two angles are called Equal if they can be placed so that their arms coincide, and that both are described simultaneously by the turning of the same line about their common vertex.
61. If two angles have the vertex and an arm in common, and do not overlap, they are called Adjacent Angles; and the angle made by the other two arms on the side toward the common arm is called the Sum of the Adjacent Angles. Thus, using the sign % for the word “angle," the sign of equality (=), and the sign of addition (+, plus),
62. A Straight Angle has its arms in the same line, and on different sides of the vertex.
63. The sum of two adjacent angles which have their exterior arms in the same line on different sides of the vertex is a straight angle.
64. When the sum of any two angles is a straight angle, each is said to be the Supplement of the other.
65. If two supplemental angles be added, their exterior arms will form one line; and then the two angles are called Supplemental Adjacent Angles.
66. A Right Angle is half a straight angle.
67. A Perpendicular to a line is a line that makes a right angle with it.
68. When the sum of two angles is a right angle, each is said to be the Complement of the other.
69. An Acute Angle is one which is less than a right angle.
70. An Obtuse Angle is one which is greater than a right angle, but less than a straight angle.
71. The whole angle which a sect must turn through, about one of its end points, to take it all around into its first position, or, in one plane, the whole amount of angle round a point, is called a Perigon.
72. Since the angular magnitude about a point is neither increased nor diminished by the number of lines which radiate from the point, the sum of all the angles about a point in a plane is a perigon.
73. A Reflex Angle is one which is greater than a straight angle, but less than a perigon.
74. Acute, obtuse, and reflex angles, in distinction from right angles, straight angles, and perigons, are called Oblique Angles; and intersecting lines which are not perpendicular to each other are called Oblique Lines.
75. When two lines intersect, a pair of angles contained by the same two lines on different sides of the vertex, having no arm in common, are called Vertical Angles.
76. That which divides a magnitude into two equal parts is said to halve or bisect the magnitude, and is called a Bisector.
77. If we imagine a figure moved, we may also suppose it to leave its outline, or Trace, in the first position.
78. A Triangle is a figure formed by three lines, each intersecting the other two.
79. The three points of intersection are the three Vertices of the triangle.
80. The three sects joining the vertices are the sides of the triangle. The side opposite A is named a; the side opposite B is 6.
81. An Interior Angle of a triangle is one between two of the sects.
82. An Exterior Angle of a triangle is one between either sect and a line which is a continuation of another side.
83. Magnitudes which are identical in every respect except the place in space where they may be, are called Congruent.
84. Two magnitudes are Equivalent which can be cut into parts congruent in pairs.
II. Properties of Distinct Things.
85. The whole is greater than its part.
87. Things which are equal to the same thing are equal to one another,