If the sides of the quadrangle are parallel two and two, the figure is called a parallelogram. FIG. 48. RECTANGLE. FIG. 49. SQUARE. If a parallelogram be right-angled, the figure is called a rectangle. If the rectangle have all its sides equal to each other, the figure is called a square. If the sides of an oblique parallelogram (none of the angles being 90°) are equal, 廾 FIG. 50. -RHOMBUS. the figure is called a rhombus. A quadrangle not having any side parallel to any other side is sometimes called a trapezium. 37. THEOREM. The sum of the exterior angles of a quadrangle is 360°, or four right angles. A point S moving in the direction indicated, about the perimeter of the quadrangle and arriving at its initial position, will have made left-handed changes of direction. at the four vertices. These changes of direction being placed adjacent to each other, as indicated in the second figure, give a complete rotation, or 360°. Q. E. D. Exercise. Show that the sum of the interior angles is four right angles. 38. THEOREM. The opposite sides of a parallelogram are equal. If the theorem be true, and we draw an auxiliary line connecting a pair of non-adjacent vertices (such a line is called a diagonal), we would have the figure separated into two triangles, which would have the three sides of one equal to the three sides of the other, and would be equal. Can we show that the triangles are equal (congruent), making use only of previously established relations? We can, because: and Z CBD = ADB, BD = BD. The two triangles have two angles and the included side in each equal, and are therefore congruent (§ 23); and hence the theorem. Exercises. 1. Use the theorem to show that parallels are everywhere equally distant from 39. THEOREM. The diagonals of a parallelogram mutually bisect each other. B FIG. 54. If the diagonals do bisect each other, the ▲ BIC and AID will be equal. Exercises. -1. Show that the diagonals of a rectangle are equal to each other. 2. Show that the diagonals of a rhombus are perpendicular to each other. 3. Show that the diagonals of a square are equal, and are perpendicular to each other. 40. Definitions. A polygon is a figure formed by a number of straight lines which enclose an area. FIG. 55. In a polygon there are as many angles as sides. A Triangle has 3 angles. A Nonagon has 9 angles. A Decagon If all the angles are equal to each other, and all the sides are equal to each other, the polygon is regular. An exterior angle of a polygon is the change of direction in going from one side to an adjacent side. Any polygon in which all the changes of direction are in the same sense, is called a convex polygon. If the changes of direction are not all in the same sense, the polygon is said to be re-entrant. 41. THEOREM. polygon is 360°. The sum of the exterior angles of any If on the perimeter of any convex polygon as repre sented in the accompanying figure we take any point as (S), and traverse the perimeter, starting in the direction indicated, and returning to (S), we shall at the vertices, A, B, C, D, and E, have made changes of direction to the left, D E S A (a) FIG. 56. amounting in all to a complete rotation, or 360°. Figure (b) represents a re-entrant polygon. As in the convex polygon, traversing the perimeter from (S), starting in the direction indicated and arriving at (S), a complete rotation will have been made. But it is to be noted that at A the change of direction is to the left, at Bit is to the right, at C, D, E, and Fit is to the left. Let P (Fig. c) be any point in the plane. From P draw lines parallel to the lines of Fig. b, which indicate the changes of direction at the succeeding vertices. |