117. Polygons are named from the number of their sides,

118. An Equilateral polygon is one all of whose sides are equal.

An Equiangular polygon is one all of whose angles are equal.

119. A polygon is called Convex when A each of its angles is less than a straight

angle; as ABCDE.

It is evident that in such a polygon no

B

E

side, if produced, can enter the space enclosed by the perimeter.

120. A polygon is called Concave when at least one of its angles is greater than a straight angle; as FGHIK, in which the interior angle whose vertex is H is greater than a straight angle.

Such an angle is called Reëntrant.

It is evident that in such a polygon at least two sides, if produced, will enter the space enclosed by the perimeter.

G

H

All polygons treated hereafter will be understood to be convex, unless the contrary is stated.

E

D

D'

E'

A

121. 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.

A

B

B'

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

Q'

gous; and angles contained by equal sides similarly 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 (91).

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 XXVI. THEOREM.

123. The sum of the interior angles of a polygon is equal to two right angles taken as many times as the polygon has sides less

Let ABCDEF be a polygon, and AE, AD, and AC diagonals. These diagonals divide the polygon into triangles.

Since the first and last triangles involve two sides of the polygon, while each other triangle only involves one side of the polygon, there will always be two triangles less than the number of sides in the polygon.

The sum of the angles of the polygon will be equal to the sum of the angles of the triangles, but (by 79) the angles of each triangle are equal to two right angles; therefore, since the number of triangles is two less than the number of sides in the polygon, the angles of the polygon are equal to two right angles taken as many times, less two, as the figure has sides.

Q.E.D.

124. COR. The sum of the angles of a quadrilateral is equal to four right angles; of a pentagon, six right angles; of a hexagon, eight right angles; etc.

125. SCHOLIUM. If R denotes a right angle, and n the number of sides of the polygon, the sum of its angles is expressed by 2 Rx (n-2), or 2 nR-4 R.

That is, the sum of the angles of a polygon is equal to twice as many right angles as the figure has sides, less four right angles.

PROPOSITION XXVII. THEOREM.

126. The exterior angles of a polygon, made by producing each of its sides in succession, are together equal to four right angles.

Let the figure ABCDE be a polygon, having its sides produced in succession.

To prove that the sum of the angles, a, b, c, d, and e are equal to four right angles.

The sum of each exterior and its corresponding interior angle (by 79) is equal to two right angles.

That is, the sum of the interior and exterior angles is equal to twice as many right angles as the figure has sides.

But by (125) the interior angles are equal to twice as many right angles as the figure has sides, less four right angles. Therefore the exterior angles alone are equal to four right angles.

Q.E.D.

EXERCISES.

1. If one side of a regular hexagon is produced, show that the exterior angle is equal to the angle of an equilateral triangle.

2. The exterior angle of a regular polygon is 18°; find the number of sides in the polygon.

3. The interior angle of a regular polygon is five-thirds of a right angle; find the number of sides in the polygon.

4. How many degrees are there in each angle of a regular pentagon ? Of a regular hexagon? Of a regular dodecagon ?

5. If two angles of a quadrilateral are supplementary, the other two angles are supplementary.

6. If a diagonal of a quadrilateral bisects two of its angles, it is perpendicular to the other diagonal.