another, then the angles were equal in magnitude but opposite in sense.

49. ASSUMPTION IV. All straight angles are equal in magnitude.

50. As a consequence, all perigons are equal in magnitude.

51. If two angles have a vertex and an arm in common, they are called adjacent angles.

FIG. 20.

52. When two adjacent angles are of the same sense, and so situated that they cannot be simultaneously described, even

in part, by the same ray rotating, their sum is the angle of like sense whose arms are their two non-coincident arms.

53. When the sum of any two angles is a straight angle, each is said to be the supplement of the other.

54. When the sum of any two angles is a perigon, each is said to be the explement of the other.

Thus ab and +ba are explemental.

FIG. 23.

FIG. 24.

α

CHAPTER II.

THE CIRCLE.

FIG. 25.

55. If, in a plane, a sect turns about one of its end points the other end point describes a curve called a circle.

56. The fixed end point is called the center of the circle.

FIG. 26.

57. Any sect from the center to a point on the curve is called a radius.

58. All radii are equal, being equal to the generating sect.

59. Since the moving sect, after rotating through a perigon, returns to its trace, therefore the moving end point describes a closed curve.

FIG. 27.

60. This curve divides the plane into two parts, one of which is finite and is swept over by the moving sect.

61. This finite plane surface is called the surface of the circle. Any point in this finite plane is said to lie within the circle.

62. Assumed Construction II. A circle may be described from any given point as center with any given sect as radius. 63. A theorem is a statement usually capable of being inferred from other statements previously accepted as true.

64. A corollary to a theorem is a statement whose truth follows readily from that of the theorem.

65. A theorem consists of two parts, the hypothesis (that which is assumed), and the conclusion (that which is asserted to follow therefrom).

66. A problem is a proposition in which something is required to be done by a process of construction.

67. The treatment of a problem in elementary geometry consists,

[1] Construction. In indicating how the ruler and compasses are to be used in effecting what is required.

[2] Proof. In showing that the construction so given is

correct.

[3] Determination. In fixing whether there is only a single solution, or suitable result of the indicated construction; or more than one; and in discussing the limitations, which sometimes exist, within which alone the solution is possible.

68. Our assumed constructions allow the use of the straightedge not marked with divisions, for drawing and producing sects, and the use of compasses for drawing circles and the transference of sects. It is important to note the implied restriction, namely, that we work in the plane, and that no construction in elementary geometry is allowable which cannot be effected by combinations of these two primary constructions.

69. Theorem. The sect to a point, from the center of a circle, is less than, equal to, or greater than the radius, according as the point is within, on, or without the circle.

Proof. For any point Q, within the circle, lies on some radius, OQR. If S is without the circle, then the sect OS contains the radius OR.

FIG. 28.

R

S

70. Inverse. A point is within, on, or without the circle,

according as its sect from the center is less than, equal to, or greater than the radius.

71. Theorem. Circles of equal radii are congruent.

Proof. For, if put in the same plane, with centers in coincidence, every point of each is on the other, because of the equality of their radii. Thus C [r] ≈ 00 [r].

72. Corollary. A circle turned about its center slides on its

trace.

This fundamental property of this curve enables us to turn any figure connected with the circle about the center without changing the relation to the circle.

73. Circles which have the same center are called concentric.. 74. Concentric circles with a point in common coincide.

75. A sect whose end points are circle is called a chord.

on the

76. Any chord through the center is called a diameter.

77. All diameters are equal, each being equal to two radii.

78. Every diameter is bisected by the center of the circle.

79. No circle can have more than one

center.

For, if it had two, the diameter through them would have two mid points.

80. Any ray from the center of a circle cuts the circle in

one, and only one, point.

81. Any straight through its center cuts the circle in two and only two points.

82. Any piece of a circle is called an arc.

83. When the end point of a radius describes an arc, the radius rotates through an angle having its vertex at the center. This angle is called the angle at the center, and is said to be subtended by the arc simultaneously described, or to stand upon that arc.

FIG. 33.

84. An arc, being described by the end point of a rotating radius, is said to have the same sense as the angle through which that radius rotates.

85. Arcs congruent and of the same sense are called equal. 86. The sum of two arcs, of the same circle, or of equal circles, is the arc which subtends an angle at the center equal to the sum of the angles subtended by those arcs separately.

87. Theorem. Equal arcs subtend equal angles at the center, and, inversely, equal angles at the center stand upon equal arcs.

Proof, For, if arc AB equal arc CD, we may slide the arc AB, together with the radii OA and OB, along the circle until A coincides with C; then will B coincide with D, since arc CD equals arc AB.

Therefore

AOB will coincide

with

B

FIG. 34.

A

FIG. 35.

COD, and will be equal to it in magnitude and sense. 88. It follows, that if A, B, C, etc., denote points on the circle and a, b, c, etc., the radii drawn to those points, then every equation between arcs AB, BC, etc., will carry with it c an equation between the corresponding angles ab, bc, etc.; and inversely.

FIG. 36.

a