It will frequently be necessary to express the magnitude of an angle by saying what part or what multiple of a right angle it is. It is important, therefore, to obtain a clear notion of the magnitude and the properties of this angle. This will be best done by considering how a right angle may be formed. We are so familiar with right angles in buildings and in articles of furniture, that we can generally judge by the eye whether a given angle is equal to, greater than, or less than a right angle. Right angles are formed by the edges of a sheet of paper, of a pane of glass, a picture-frame, a door. We may readily make a right angle thus: Take a sheet of writing-paper, fold it in two; the crease will form a straight line, A B; fold it again, so that one part, C B A FIG. 23. B A OB, of the line falls on the other part, O A (Fig. 23); the sheet, now folded into four, will have the two edges, A O, CO, perpendicular to one another. A portable right angle, like the one here formed, is technically termed a set-square. The Set-square.—The draughtsman's set-square (Fig. 24) is a thin piece of board, having an angle exactly equal to the angle of the paper model described above. By means of this instrument a right angle may be drawn on paper in any position. For example, let it be required to draw from the point M in the straight line, A B, a perpendicular to A B. Place a flat ruler so that its edge lies along the line A B, and then place the set-square in FIG. 24. contact with the ruler, and so that the apex of the right angle coin cides with the point M, as in the figure M FIG. 25 B In constructing set-squares and the other instruments used for similar purposes, which we are about to describe, great care must be taken to make the right angle true at first, and to fix the parts so that it may remain true. To test a set-square, first see that the edges are quite straight; then apply the square to a ruler, and draw straight line E E FIG. 26. FIG. 27. perpendicular to the ruler. Next, without moving the ruler, turn the square over this line, and apply it again to the edge of the ruler, so that it may pass from its position E to E'. If the edge of the square coincide in this new position with the line, it is true. Fig. 26 represents the test of an untrue square, each of the angles being less than a right angle. Figure 27 represents the set-square ordinarily used by carpenters and joiners. T is termed the stock and L the blade. The T-Square. Another kind of set-square, called, from its shape, a T-square, is used for making lines per pendicular to the edge of a drawing - board (Fig. 28). FIG. 28. The Bevel. The in struments we have described have the parts fixed so that the angle may remain invariable; but artisans also require a means of making angles of different sizes. FIG. 29. For this purpose a bevel is used. It consists of a blade and stock, similar to the parts of the carpenter's setsquare, which turn upon an axis, like the legs of a pair of compasses, so as to form an angle of any size (Fig. 29). Horizontal and Vertical Lines.-We have already explained that a vertical line at any point on the earth's surface is the direction of the plumb-line at that point, and that a horizontal line is a line lying in a horizontal plane, or plane touching the surface of still water. Every horizontal line passing through a point is perpendicular to the vertical line through the same point, and every line drawn at right FIG. 30. with a plumb-line attached. a level. FIG. 31. angles to a vertical line is a horizontal line. If we place one of the sides of a setsquare against the plumb-line, the other will be horizontal (Fig. 30). The side-posts of a door are vertical, and therefore the lintel and bottom of the door, being fixed at right angles to the sideposts, are horizontal. A horizontal and a vertical line form with each other a right angle in a particular position with regard to the earth's surface. In the erection of buildings the workmen have frequently to test the lines of their work to see that they are horizontal or vertical. The vertical lines are easily tested by the plumb-line, and the hori zontal lines are tested by means of a large set-square Such an instrument is usually termed middle board, and a plumb-line is attached to a point, C, in this line. When the edge A B is horizontal, the plumb-line covers the marked line CD. Fig. 32 represents a mason's level, used to see whether two points, A and B, are in the same horizontal line; the point A is not at the same level as the point B-that is to say, A and B are not in the same horizontal line; consequently the plumb-line does not cover the marked line CD, but lies to the left of it, because A FIG. 32. A, on the left, is lower than B. The mason's level requires to be frequently tested, as the wood warps and wears unequally. It is tested by placing it on two points, A and B, and altering the height of these points until the plumb-line lies on the marked line, then reversing the instrument, so that the end which was on A falls on B. If the instrument be true, the plumb-line will still fall on the marked line. Propositions on Angles.-If from a point C in a straight line, A B, another straight line, CD. be drawn. so as to make two adjacent angles, D C B, DC A, the two angles will be together equal to two right angles (Fig. 33). E A B FIG. 33. If DC cut the straight line A B in E, we shall have at the point E two angles, DEB, AEC, on opposite sides of the common vertex, E; such that the sides of one are the continuation of the sides of the other. Two such angles are called vertically-opposite angles. Vertically-opposite angles are equal (Fig. 34). The two statements here made admit of proof, that is to say, it can be shown, by means of the definitions and the properties of lines and angles, previously established, that they must be true. The proofs of these and other propositions must be remembered in the words of the book, or in words equivalent to them; and as it is very important that A FIG. 34 B the student shall have at every step a clear knowledge of what has been previously proved, we have separated the definitions and propo sitions, with their proofs, from the merely explanatory matter, and printed them in larger type, and have also numbered them to facilitate reference. It will be seen that the statement of what is to be proved is made twice over, viz., first in general terms, and next, with reference to a particular figure. Easy Exercises. (a). Make for yourself a set-square of paper, like that represented in Fig. 21. (b). Draw a straight line, A B, and then with your set-square draw a line, A C, perpendicular to A B. (c). Mention any objects which have right angles on their surfaces, one side of each right angle being horizontal. (d). Mention any objects which have right angles on their surfaces. neither of the sides being horizontal. (e). Take a point on a sheet of paper, and draw from it two straight lines which shall contain an acute angle. (f). Draw two straight lines which shall contain an obtuse angle. (g). Make two angles which shall be supplementary. (h). If from a point on the ground, straight lines be drawn in directions north, south, east, and west, what will be the magnitude of each angle formed, and what will be the sum of the whole ? ABBREVIATIONS. Throughout this work the following signs are used: for the words "the angle." "the triangle." "is equal to." "is greater than." C |