POLYGONS IN GENERAL. 192. A polygon is a plane figure bounded by straight lines. The bounding lines are the sides of the polygon, and 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 their vertices are the vertices of the polygon. The number of sides of a polygon is evidently equal to the number of its angles. 193. A diagonal of a polygon is a line joining the vertices of two angles not adjacent; as AC, Fig. 1. 194. An equilateral polygon is a polygon which has all its sides equal. 195. An equiangular polygon is a polygon which has all its angles equal. 196. A convex polygon is a polygon of which no side, when produced, will enter the surface bounded by the perimeter. 197. Each angle of such a polygon is called a salient angle, and is less than a straight angle. 198. A concave polygon is a polygon of which two or more sides, when produced, will enter the surface bounded by the perimeter. Fig. 3. 199. The angle FDE is called a re-entrant angle, and is greater than a straight angle. If the term polygon is used, a convex polygon is meant. 200. 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. 201. Two polygons are mutually equiangular, if the angles of the one are equal to the angles of the other, each to each, when taken in the same order. Figs. 1 and 2. 202. The equal angles in mutually equiangular polygons are called homologous angles; and the sides which lie between equal angles are called homologous sides. 203. Two polygons are mutually equilateral, if the sides of the one are equal to the sides of the other, each to each, when taken in the same order. Figs. 1 and 2. Two polygons may be mutually equiangular without being mutually equilateral; as, Figs. 4 and 5. And, except in the case of triangles, two polygons may be mutually equilateral without being mutually equiangular; as, Figs. 6 and 7. If two polygons are mutually equilateral and equiangular, they are equal, for they may be applied the one to the other so as to coincide. 204. A polygon of three sides is called a trigon or triangle; one of four sides, a tetragon or quadrilateral; one of five sides, a pentagon; one of six sides, a hexagon; one of seven sides, a heptagon; one of eight sides, an octagon; one of ten sides, a decagon; one of twelve sides, a dodecagon. PROPOSITION XLIII. THEOREM. 205. 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. Let the figure ABCDEF be a polygon having n sides. To prove ZA+ZB+ZC, etc. (n-2) 2 rt. 4. Proof. From the vertex A draw the diagonals AC, AD, and AE. The sum of the of the A polygon. = the sum of the s of the Now there are (n − 2) ▲, and the sum of the s of each A 2 rt. s. § 138 .. the sum of the s of the A, that is, the sum of the sof the polygon = (n − 2) 2 rt. ≤. Q. E. D. 206. COR. 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 are all equal, each angle is a right angle. In general, each angle of an equiangular polygon of n sides is 2(n-2) equal to right angles. n PROPOSITION XLIV. THEOREM. 207. 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 the sum of the ext. = 4 rt. s. Proof. Denote the int. and the ext. of the polygon by A, B, C, D, E, by a, b, c, d, e. ZA+Za=2 rt. 4, $ 90 and ZB+2b=2 rt. 4, (being sup.-adj. 4). = 2 rt. 4. In like manner each pair of adj. .. the sum of the interior and exterior 2 rt. taken as many times as the figure has sides, or, = 2 n rt. . But the interior 2 rt. 4 taken as many times as the figure has sides less two, = (n-2) 2 rt. 4, or, 2 n rt. 4 rt. . .. the exterior 4 rt. . Q. E. D. PROPOSITION XLV. THEOREM. 208. A quadrilateral which has two adjacent sides equal, and the other two sides equal, is symmetrical with respect to the diagonal joining the vertices of the angles formed by the equal sides, and the diagonals intersect at right angles. Let ABCD be a quadrilateral, having AB = AD, and CB= CD, and having the diagonals AC and BD. To prove that the diagonal AC is an axis of symmetry, and is to the diagonal BD. Proof. In the A ABC and ADC (having three sides of the one equal to three sides of the other). :. Z BAC=Z DAC, and ▲ BCA = 2 DCA, .. if ABC is turned over on AC as an axis, AB will fall upon AD, CB on CD, and OB on OD. Hence AC is an axis of symmetry, § 65, and is to BD. Q. E. D. |