SECTION V. POLYGONS

69. Polygons. The limited portion of a plane bounded by a broken line which is closed (§ 26) is called a polygon. The amount of surface within the polygon is called the area of the polygon, and the length of the boundary is called the perimeter of the polygon. The term "perimeter" is often used for the broken line bounding the polygon when no idea of length is involved, but when this is done, the context usually makes clear which meaning is to be attached to the word.

70. Vertices and Sides. The vertices of the angles formed by the sects of the broken line are called the vertices of the polygon, and the sects themselves are called the sides of the polygon. The figure ABCDE is a polygon.

which are marked with the arrows are interior angles.

The angle at any vertex of a polygon having as arms one side of the polygon, and the continuation of another side, is called an exterior angle, as ▲ XBC. In speaking

of the exterior angles of a polygon, the angles formed by producing the sides in succession, each through the vertex formed with the following one, in passing around the polygon, are meant. There are evidently two sets of exterior angles of a polygon, formed by passing around clockwise or counterclockwise. These sets are, however, equal, for the two exterior angles at any vertex are vertical.

72. Diagonals. A line joining two non-consecutive vertices of a polygon is called a diagonal.

73. Concave, Convex, and Cross Polygons. A polygon is said to be convex if no side when produced could cut the surface of the polygon. Unless otherwise stated, convex polygon will be meant whenever the term "polygon" is used.

A polygon is concave when at least one side, if produced, would cut its surface; it is called cross when its perimeter intersects itself.

74. Equilateral, Equiangular, and Regular Polygons. A polygon that has all its sides equal is called equilateral; one that has all its angles equal is called equiangular. A polygon that is both equilateral and equiangular is called regular.

75. Number of Sides. A polygon of three, four, five, six, eight, ten sides is called, respectively, a triangle, quadrilateral, pentagon, hexagon, octagon, decagon, etc.

76. Base. The side of to stand is called its base. be considered the base.

a polygon on which it appears Any side of a polygon might.

TRIANGLES

77. Parts of a Triangle. A triangle has six parts, three sides and three angles. An angle and a side are spoken of as opposite to each other when the side is not one of the arms of the angle. A side is sometimes spoken of as included between two angles, and an angle as included between two sides, when the order in which they lie is

meant.

78. Vertex Angle. The angle opposite the base of a triangle is called the vertex angle, or the vertical angle.

79. Equality of Sides. If a triangle has two equal sides, it is called isosceles; if no equal sides, scalene. In an isosceles triangle, the equal sides are sometimes called legs, the third side the base.

80. Angles of a Triangle. If all the angles of a triangle are acute, it is called an acute-angled triangle; if one angle is right, it is called a right triangle; and if one angle is obtuse, it is called an obtuse-angled triangle. In a right triangle the side opposite the right angle is called the hypotenuse, and the other sides the legs.

81. Lines of a Triangle. There are four kinds of lines of importance in work with a triangle: the bisectors of the angles, the perpendicular bisectors of the sides, the altitudes, and the medians. The first two explain themselves; an altitude is a perpendicular from a vertex to the opposite side, and a median is a line from a vertex to the midpoint of the opposite side. If the altitude of a triangle is spoken of, the altitude to the base is meant.

SECTION VI. INEQUALITIES

82. Axiom of Unequals. The whole is greater than any of its parts.

83. Inequalities. There are certain truths relating to statements of inequality that depend very closely on the general axioms. Their proofs are given under the head of Inequalities in nearly all Algebras, so they will not be considered here. They are used for those magnitudes for which equality axioms 2-5 are used.

(1) If equals are added to, taken from, multiplied by, or divided into, unequals, the results are unequal in the

same sense.

(That is, the greater quantity remains greater after the operation is performed.)

(2) If unequals are taken from, or divided into equals, the results are equal in the opposite sense.

(3) If unequals are added to, or multiplied by, unequals in the same sense, the results are unequal in the

same sense.

(4) If the first of several magnitudes is greater than the second, the second greater than the third, the third greater than the fourth, and so on, then the first is greater

than the last.

NOTE. (4) holds for "the first less than the second," etc.; also any pairs might be equal without changing the result, if there is at least one inequality.

These statements are all in regard to positive magnitudes; they are not all true when negative quantities are used.

WARNING. Unequals should not be taken from, or divided into, unequals, for the results cannot, in general, be determined.

SECTION VII. PROPOSITIONS

84. Propositions. Proposition is a general term including:

(1) Theorem, which is a truth to be proved.

(2) Corollary, which is also a truth to be proved, but generally one that follows quite directly, and often very simply, from a known truth-most often from a theorem that has just been proved.

(3) Problem, or Construction Theorem, which requires that a certain figure be drawn from given parts. A more definite understanding of problem will be given in § 177.

85. Parts of a Theorem. A theorem is composed of the condition, or hypothesis, and the conclusion. It usually

takes the form, If a certain condition is true, a certain conclusion is also true. (See § 2.)

86. Proof of a Proposition. To prove any proposition, the student has certain materials from which to work, namely the foregoing definitions, axioms, and truths concerning them; the theorems preceding the one that is being proved; and the condition of the proposition in question. From these known truths the proof must be deduced, and the required conclusion must be reached.

The proof must always be a logical one; the truth of a proposition must not be judged by measurements—as in Concrete Geometry or from the appearance of the figure, for such methods have no place in this subject. It might be said, however, that a carefully drawn figure will some