THEOREM XVIII. The exterior angle of any triangle is equal to the two interior and opposite angles together, and the three angles of a triangle are together equal to two right angles. Let A B C be a triangle of which A C D is the exterior angle; then shall C E 4 ACD = 4 CAB+ 4ABC Suppose CE to be a line drawn from C parallel to B A, sponding angles, And ECA 4 CAB because they are alternate angles. Therefore ACD=4CAB+ ZABC; Consequently, also, the three angles CA B, ABC, BCA are together equal to the two adjacent angles AC D, A C B, and therefore to two right angles. THEOREMS FOR EXERCISE. 22. Two straight lines which are perpendicular to a third are parallel. 23. A perpendicular to one of two parallels is also perpendicular to the other. 24. If two lines, one of which is oblique and the other perpendicular to the same straight line, be produced far enough, they must necessarily intersect. 25. If two straight lines be parallel to two other straight lines which intersect, these two straight lines shall also intersect and form at the point of intersection four angles equal respectively to the angles formed by the other straight lines. 26. In a right-angled triangle, if one acute angle be twice the other, the hypothenuse shall be double of the shorter side. 27. In any triangle, A B C, two lines, A D, A E, are drawn to BC, making ▲ B A D = ≤ C and ▲ CAE ▲ B. Prove that A E D is an isosceles triangle. = 28. Two straight lines, perpendicular to two other straight lines which intersect, cut one another and form at the point of intersection four angles equal respectively to the angles made by the other straight lines. 29. Having drawn a line from the vertex to the middle point of the base of a triangle, prove that when this middle line is greater than half the base the vertical angle is acute ; when the middle line is equal to half the base the angle is a right angle; and when the middle line is less than half the base the vertical angle is obtuse. CHAPTER VI. QUADRILATERALS Definitions. We have now to consider the properties of four-sided have to consider the relations between the side, angles, and diagonals of certain kinds of quadrilaterals. The quadrilateral represented in Fig. 716, having two sides parallel, is termed a trapezoid. Figs. 72a, b, c, d, are quadrilaterals, which are also termed parallelograms, because in each the opposite sides are parallel. Fig. 72c is a parallelogram, having all its angles right angles, and is therefore termed a rectangle. A rectangle with equal sides is termed a square (Fig. 72d). Relations of the Figures to one another.-Arranging these in order of generality-that is to say, so that each term includes all that follow-we have 1. Quadrilateral. 2. Trapezoid. 3. Parallelogram. 4. Rectangle. 5. Square. Every property belonging to any one of these figures belongs also to all that follow, but not necessarily to all that precede. For instance, it is true of all that the sum of the angles is four right angles; it is true of 2, 3, 4 and 5 that the angles may be divided into two pairs of supplementary angles, but this is not true of every quadrilateral; it is true of 3, 4, and 5 that the opposite angles are equal, but there may be quadrilaterals and trapezoids of which this is not true; similarly, it is necessarily true only of 4 and 5 that the diagonals bisect one another and are equal, and of 5 only that the equal diagonals bisect one another at right angles. Applications. All these figures have many important applica tions. (a). In Construction.—A symmetrical trapezoid has frequently to be constructed by carpenters. It is the remainder of an isosceles triangle when a portion has been cut off by a line parallel with the base. The trussed beam with an upper and lower tie-beam affords an example of this figure. FIG. 73. The parts of a trussed beam are united by what are called dovetailed joints (Fig. 74), to prevent the parts from being drawn asunder by the strain; these joints are symmetrical trapezoids. The cavity is termed the mortise, the projection a b c d the tenon. The two parts are fitted by laying the tie-beam A upon the upright B, and hammering the tenon into the mortise. It is evident that the outward strain will only make the joint tighter. Fig. 75 represents a compound dovetailed joint. Picture-frames, panels, and B FIG. 74. the covers of boxes are often bounded by symmetrical trapezoids Fig. 76. (b). In the Theory of Forces.-A very important use of the parallelogram is found in the theory of forces. If two forces act on a body, and two straight lines be drawn parallel to the forces, and having lengths which represent the magnitudes of the forces, then the single force which will have the same effect as the two forces will be that represented by the diagonal of the parallelogram which can be made on the two straight lines. For instance, suppose the forces to be 3 lbs. and 2 lbs., and let OB be parallel to the direction of the force of 3 lbs, and OA to that of 2 lbs. Let three times a certain length be marked off B FIG. 77 C along O B and twice this length along O A. Let B C be drawn parallel to O A, and let AC be parallel to OB, so that O ACB shall be a parallelogram of which O C is a diagonal. Then OC represents the single force which would have the same effect as the two forces together. Suppose, for example, OC contains four times the length taken to represent I lb., then 3 lbs. along BO and 2 lbs. along A O produce the same effect as 4 lbs. along CO. If then we continue CO to C', 4 lbs. along C'O would exactly balance the two forces, 3 lbs. along B O and 2 lbs. along A O, and the body at O would remain at rest when all three forces are acting together. (c). The Parallel Ruler.-It will be proved that the opposite sides of a parallelogram are equal, and also that when the opposite sides of a four-sided figure are equal the figure is a parallelogram. This shows that if we make a parallelogram with four rods jointed at the corners, so that its shape may be altered, the opposite sides, being equal at first, will always be equal; so that the figure formed by the rods will always be a parallelogram. The parallel ruler is based on this fact. (d). In the Steam Engine of Watt.-Watt, whose name is famous on account of his inventions connected with the steam engine, used the parallelogram as a means of producing the oscillation of the balance |