POLYGONS. DEFINITIONS. 107. A Polygon is a plane figure bounded by straight lines. The bounding lines are the sides of the polygon. The Perimeter of a polygon is the sum of the bounding lines. The angles which the adjacent sides make with each other are the angles of the polygon. A Diagonal of a polygon is a line joining two non-adjacent angles. NOTE. Let the pupil illustrate. 108. An Equilateral Polygon is one all of whose sides are equal. 109. An Equiangular Polygon is one all of whose angles are equal. Two polygons are mutually equilateral when their sides are respectively equal. Two polygons are mutually equiangular when their angles are respectively equal. Homologous sides or angles are those which are similarly placed. 110. A Convex Polygon is one no side of which when produced can enter the surface bounded by the perimeter. Each angle of such a polygon is called a salient angle. 111. A Concave Polygon is one of which two or more sides, when produced, will enter the space enclosed by the perimeter. The angle AOC is called a re-entrant angle. 112. By drawing diagonals from the vertex of any angle of a polygon, it may be divided into as many triangles as it has sides less two. 2 3 113. A polygon of three sides is a Triangle; of four, a Quadrilateral; of five, a Pentagon; of six, a Hexagon; of seven, a Heptagon; of eight, an Octagon; of nine, a Nonagon; of ten, a Decagon; of twelve, a Dodecagon. ANGLES OF A POLYGON. THEOREM XXXVIII. 114. The sum of all the angles of any polygon equals two right angles taken as many times less two as the polygon has sides. To prove that AFE + ▲ FED + ≤ EDC, etc. = 2 Ls (n-2) From any vertex, as F, draw the diagonals FB, FC, and FD. Then we have (n−2) ▲s. (112) The sum of the Ls of the As = the sum of the sof the polygon. But the sum of the Ls of a ▲ = 2 Ls; (76) Q. E. D. the sum of the Zs of the polygon =2 Ls (n-2). 115. COR.-The sum of the angles of a quadrilateral is 4 Ls; of a pentagon, 6 Ls; of a hexagon, 8 Ls, etc. THEOREM XXXIX. 116. If the sides of a convex polygon are produced so as to form one exterior angle at each vertex, the sum of the exterior angles equals four right angles. Let ABCDE be a polygon of n sides, and let the sides be produced so as to form the exterior s a, b, c, d, e. To prove that a + 2 b + 2 c + L d + Le = 4 Ls. At each vertex there are two s whose sum = 2 Ls; (44) and since there are as many vertices as there are sides, we have n 2 Ls. But the sum of the interior s = 2 Ls (n − 2); ...La+<b+2c+<d+ Le = n × 2 Ls = = (114) 2 Ls (n-2) 2nLs-2n Ls + 4 Ls. = 4 Ls. Q. E. D. QUADRILATERALS. DEFINITIONS. 117. A Quadrilateral is a polygon of four sides. 118. There are three classes of quadrilaterals, namely, Trapeziums, Trapezoids, and Parallelograms. 119. A Trapezium is a quadrilateral which has no two of its sides parallel. 120. A Trapezoid is a quadrilateral which has two of its sides parallel. The parallel sides are called the bases. 121. A Parallelogram is a quadrilateral which has its opposite sides parallel. The side upon which a parallelogram is supposed to stand and the opposite side are called the bases. 122. A Rectangle is a parallelogram whose angles are right angles. 123. A square is an equilateral rectangle. 124. A Rhomboid is a parallelogram whose angles are oblique. |