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152. Def. Two polygons are Equal, when they can be divided by diagonals into the same number of triangles, equal each to each, and similarly placed ; for the polygons can be applied to each other, and the corresponding triangles will evidently coincide. Therefore the polygons will coincide, and be equal in all respects.

153. Def. Two polygons are Mutually Equiangular, if the angles of the one be equal to the angles of the other, each to each, when taken in the same order; as the polygons ABC DE F, and A' B' C' D' E' F', in which <A = L A', B= B', ZC= _ C', etc.

154. DEF. The equal angles in mutually equiangular polygons are called Homologous angles; and the sides which lie between equal angles are called Homologous sides.

155. DEF. Two polygons are Mutually Equilateral, if the sides of the one be equal to the sides of the other, each to each, when taken in the same order.

Fig. 4.

Fig. 1.

Fig. 2.

Fig. 3. Two polygons may be mutually equiangular without being mutually equilateral ; as Figs. 1 and 2.

And, except in the case of triangles, two polygons may be mutually equilateral without being mutually equiangular; as Figs. 3 and 4.

If two polygons be mutually equilateral and equiangular, they are equal, for they may be applied the one to the other so as to coincide.

156. DEF. 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 sides is an Octagon ; one of ten sides is a Decagon ; one of twelve sides is a Dodecagon.

ON POLYGONS IN GENERAL.

143. DEF. A Polygon is a plane figure bounded by straight lines.

144. DEF. The bounding lines are the sides of the polygon, and their sum, as A B + B C + C D, etc., is the Perimeter of the polygon.

The angles which the adjacent sides make with each other are the angles of the polygon.

145. DEF. A Diagonal of a polygon is a line joining the vertices of two angles not adjacent.

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146. DEF. An Equilateral polygon is one which has all its sides equal.

147. DEF. An Equiangular polygon is one which has all its angles equal.

148. DEF. A Convex polygon is one of which no side, when produced, will enter the surface bounded by the perimeter.

149. Def. Each angle of such a polygon is called a Salient angle, and is less than two right angles.

150. DEF. A Concave polygon is one of which two or more sides, when produced, will enter the surface bounded by the perimeter.

151. DEF. The angle FD E is called a Re-entrant angle. When the term polygon is used, a convex polygon is meant.

The number of sides of a polygon is evidently equal to the number of its angles.

By drawing diagonals from any vertex of a polygon, the figure may be divided into as many triangles as it has sides less two. 152. Def. Two polygons are Equal, when they can be divided by diagonals into the same number of triangles, equal each to each, and similarly placed; for the polygons can be applied to each other, and the corresponding triangles will evidently coincide. Therefore the polygons will coincide, and be equal in all respects.

153. Def. Two polygons are Mutually Equiangular, if the angles of the one be equal to the angles of the other, each to each, when taken in the same order; as the polygons ABC DE F, and A' B' C' D' E' F', in which Z A = Z A', Z B= 2 B', ZC= _ C', etc.

154. DEF. The equal angles in mutually equiangular polygons are called Homologous angles; and the sides which lie between equal angles are called Homologous sides.

155. DEF. Two polygons are Mutually Equilateral, if the sides of the one be equal to the sides of the other, each to each, when taken in the same order.

Fig. 1.
Fig. 2.
Fig. 3.

Fig. 4. Two polygons may be mutually equiangular without being mutually equilateral ; as Figs. 1 and 2.

And, except in the case of triangles, two polygons may be mutually equilateral without being mutually equiangular; as Figs. 3 and 4.

If two polygons he mutually equilateral and equiangular, they are equal, for they may be applied the one to the other so as to coincide.

156. DEF. 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 Heptayon; one of eight sides is an Octagon ; one of ten sides is a Decagon ; one of twelve sides is a Dodecagon.

Proposition XLVI. THEOREM. 157. The sum of the interior angles of a polygon is equal to two right angles, taken as many times less two as the figure has sides.

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Let the figure ABCDEF be a polygon having n sides.

We are to prove

ZA + B + 2 C, etc., = 2 rt. 4 (n − 2).
From the vertex A draw the diagonals A C, A D, and A E.

The sum of the ts of the A = the sum of the angles of the polygon.

Now there are (n — 2) A and the sum of the 6 of each A = 2 rt. I. $ 98 .. the sum of the ts of the A, that is, the sum of the Is of the polygon = 2 rt. (n — 2).

Q. E. D. 158. COROLLARY. The sum of the angles of a quadrilateral equals two right angles taken (4 — 2) times, i. e. equals 4 right angles; and if the angles be all equal, each angle is a right angle. In general, each angle of an equiangular polygon of n sides is equal to 2 (n. — 2) richt

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2 (n. — 2) right angles.

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