=

and BAD CAD; that is, the straight line joining the vertex and the middle of the base of an isosceles triangle is perpendicular to the base and bisects the vertical angle.

Hence, also, the straight line which bisects the vertical angle of an isosceles triangle bisects the base at right angles.

88. Corollary II. Every equilateral triangle is also equiangular; and by (68), each of its angles is equal to one-third of two right angles, or to two-thirds of one right angle.

PROPOSITION XXVI.-THEOREM.

89. If two sides of a triangle are unequal, the angles opposite to them are unequal, and the greater angle is opposite to the greater side. In the triangle ABC, let AB be greater than AC; then, the angle ACB is greater than the angle B.

B

D

Α

For, from the greater side AB cut off a part AD= AC, and join CD. The triangle ADC is isosceles, and therefore the angles ADC and ACD are equal (86). But the whole angle ACB is greater than its part ACD, and therefore greater than ADC; and ADC, an exterior angle of the triangle BDC, is greater than the angle B (69); still more, then, is ACB greater than B.

PROPOSITION XXVII.-THEOREM.

90. If two angles of a triangle are equal, the sides opposite to them are equal.

In the triangle ABC, let the angles B and C be

equal; then, the sides AB and AC are equal.

А

For, if the sides AB and AC were unequal, the angles B and C could not be equal (89).

B

91. Corollary. Every equiangular triangle is also equilateral.

92. If two angles of a triangle are unequal, the sides opposite to them are unequal, and the greater side is opposite to the greater angle. In the triangle ABC let the angle C be greater than

the angle B; then, AB is greater than AC.

For, suppose the line CD to be drawn, cutting off from the greater angle a part BCD

B.

Then BDC
But in the

is an isosceles triangle, and DC= DB.

triangle ADC, we have AD + DC > AC; or, putting DB for its equal DC, AD + DB > AC; or AB > AC.

B

A

POLYGONS.

B

A

E

93. Definitions. A polygon is a portion of a plane bounded by straight lines; as ABCDE. The bounding lines are the sides; their sum is the perimeter of the polygon. The angles which the adjacent sides make with each other are the angles of the polygon; and the vertices of these angles are called the vertices of the polygon.

Any line joining two vertices not consecutive is called a diagonal; as AC.

94. Definitions. Polygons are classed according to the number of their sides:

A triangle is a polygon of three sides.

A quadrilateral is a polygon of four sides.

A pentagon has five sides; a hexagon, six; a heptagon, seven; an octagon, eight; an enneagon, nine; a decagon, ten; a dodecagon, twelve; etc.

An equilateral polygon is one all of whose sides are equal; an equiangular polygon, one all of whose angles are equal.

95. Definition. A convex polygon is one no side of which when produced can enter within the space enclosed by the perimeter, as ABCDE in (93). Each of the angles of such a polygon is less than two right angles.

It is also evident from the definition that the perimeter of a convex

polygon cannot be intersected by a straight line in more than two points.

M

(

A concave polygon is one of which two or more sides, when produced, will enter the space enclosed by the perimeter; as MNOPQ, of which OP and QP when produced will enter within the polygon. The angle OPQ, formed by two adjacent re-entrant sides, is called a reentrant angle; and hence a concave polygon is sometimes called a re-entrant polygon.

N

All the polygons hereafter considered will be understood to be

convex.

D

E

96. A polygon may be divided into triangles by diagonals drawn from one of its vertices. Thus the pentagon ABCDE is divided into three triangles by the diagonals drawn from A. The number of triangles into which any polygon can thus be divided is evidently equal to the number of its sides, less two. The number of diagonals so drawn is equal to the number of sides, less three.

A

E

97. Two polygons ABCDE, A'B'C'D'E', are equal when they can be divided by diagonals into the same number of triangles, equal each to each, and similarly arranged; for the polygons can evidently be superposed, one upon the other, so as to coincide.

98. Definitions. Two polygons

are mutually equiangular when the angles of the one are respectively equal to the angles of the other, taken in the same order; as ABCD, A'B'C'D', in which A A', B

=

=

B', etc.

A

B

D

B

C

B

The equal angles are called homologous angles; the sides containing equal angles, and similarly placed, are homologous sides; thus A and A' are homologous angles, AB and A'B' are homologous sides, etc.

Two polygons are mutually equilateral when the sides of the one are respectively equal to the sides of the other, taken in the same order; as MNPQ, M'N'P'Q', in which

MN M'N', NP = N'P', etc.

=

The equal sides are homologous; and angles contained by equal sides simi

M'

M

P

N'

larly placed, are homologous; thus MN and M'N' are homologous sides; M and M' are homologous angles.

Two mutually equiangular polygons are not necessarily also mutually equilateral. Nor are two mutually equilateral polygons necessarily also mutually equiangular, except in the case of triangles (80).

If two polygons are mutually equilateral and also mutually equiangular, they are equal; for they can evidently be superposed, one upon the other, so as to coincide.

PROPOSITION XXIX.-THEOREM.

99. The sum of all the angles of any polygon is equal to two right angles taken as many times less two as the polygon has sides.

For, by drawing diagonals from any one vertex, the polygon can be divided into as many triangles as it has sides, less two (96). The sum of the angles of all the triangles is the same as the sum of the angles of the polygon, and the sum of the angles of each triangle is two right angles (68). Therefore, the sum of the angles of the polygon is two right angles taken as many times less two as the polygon has sides.

100. Corollary I. If N denotes the number of the sides of the polygon, and R a right angle, the sum of the angles is 2R × (N-2) (2N — 4) R = 2NR 4R; that is, twice as many right angles as the polygon has sides, less four right angles.

=

For example, the sum of the angles of a quadrilateral is four right angles; of a pentagon, six right angles; of a hexagon, eight right angles, etc.

4

101. Corollary II. If all the sides of any polygon ABCDE, be produced so as to form one exterior angle at each vertex, the sum of these exterior angles,

e/E

D

A

B
b

a, b, c, d, e, is four right angles. For, the sum of each interior and its adjacent exterior angle, as Aa, is two right angles (11); therefore, the sum of all the angles, both interior and exterior, is twice as many right angles as the polygon has sides. But the sum of the interior angles alone is twice as many right angles as the polygon has sides; less four right angles (100); therefore the sum of the exterior angles is equal to four right angles.

This is also proved in a very simple manner, by drawing, from any point in the plane of the polygon, a series of lines respectively parallel to the sides of the polygon and in the same directions as their prolongations. The consecutive angles formed by these lines will be equal to the exterior angles of the polygon (60), and their sum is four right angles (15).

QUADRILATERALS.

102. Definitions. Quadrilaterals are divided into classes as follows:

1st. The trapezium (A) which has no two of its sides parallel.

2d. The trapezoid (B) which has two sides parallel. The parallel sides are called the bases, and the perpendicular distance between them the altitude of the trapezoid.

3d. The parallelogram (C) which is bounded by two pairs of parallel sides.

The side upon which a parallelogram is supposed

B

C

A

to stand and the opposite side are called its lower and upper bases. The perpendicular distance between the bases is the altitude.

103. Definitions. Parallelograms are divided into species, as follows