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CHAPTER VIII.

POLYGONS.

I. Definitions.

192. A Polygon is a figure formed by a number of lines of which each is cut by the following one, and the last by the first.

193. The common points of the consecutive lines are called the Vertices of the polygon.

194. The sects between the consecutive vertices are called the Sides of the polygon.

195. The sum of the sides, a broken line, makes the Perimeter of the polygon.

196. The angles between the consecutive sides and towards the enclosed surface are called the Interior Angles of the polygon. Every polygon has as many interior angles as sides.

197. A polygon is said to be Convex when no one of its interior angles is reflex.

198. The sects joining the vertices not consecutive are called Diagonals of the polygon.

199. When the sides of a polygon are all equal to one another, it is called Equilateral.

200. When the angles of a polygon are all equal to one another, it is called Equiangular.

201. A polygon which is both equilateral and equiangular is called Regular.

202. Two polygons are Mutually Equilateral if the sides of the one are equal respectively to the sides of the other taken in the same order.

203. Two polygons are Mutually Equiangular if the angles of the one are equal respectively to the angles of the other taken in the same order.

204. Two polygons may be mutually equiangular without being mutually equilateral.

205. Except in the case of triangles, two polygons may be mutually equilateral without being mutually equiangular.

206. A polygon of three sides is a Trigon or Triangle; one of four sides is a Tetragon or Quadrilateral; one of five sides is a Pentagon; one of six sides is a Hexagon; one of seven sides is a Heptagon; one of eight, an Octagon ; of nine, a Nonagon; of ten, a Decagon; of twelve, a Dodecagon; of fifteen, a Quindecagon.

207. The Surface of a polygon is that part of the plane enclosed by its perimeter.

208. A Parallelogram is a quadrilateral whose opposite sides are parallel.

209. A Trapezoid is a quadrilateral with two sides parallel.

II. General Properties.

THEOREM XXVIII.

210. If two polygons be mutually equilateral and mutually equiangular, they are congruent.

Proof. Superposition: they may be applied, the one to the other, so as to coincide.

How

EXERCISES. 29. Is a parallelogram a trapezoid ? could a triangle be considered a trapezoid ?

THEOREM XXIX.

211. The sum of the interior angles of a polygon is two less straight angles than it has sides.

HYPOTHESIS. A polygon of n sides.
CONCLUSION. Sum of x's (n 2) st. X's.

PROOF. If we can draw all the diagonals from any one vertex with out cutting the perimeter, then we have a triangle for every side of the polygon, except the two which make our chosen vertex. Thus, we have (n 2) triangles, whose angles make the interior angles of the polygon.

But, by 174, the sum of the angles in each triangle is a straight angle,

Sum of X's in polygon (n 2) st. X's.

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212. COROLLARY I. From each vertex of a polygon of n sides are (12

3) diagonals. 213. COROLLARY II. The sum of the angles in a quadrilateral is a perigon.

214. COROLLARY III. Each angle of an equiangular poly

(n − 2) st. *'s gon of n sides is

n

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