(2) If two sides of a triangle are unequal, the opposite angles are unequal, the larger angle being opposite the longer side. Then, since the conditions with regard to the sides cover all possibilities, and only one of the conclusions can be true, the converses of these statements must follow. Stated with regard to triangle ABC, Given; If BCCA, then correspondingly A≥≤B. It follows that if ≤ 4B, then correspondingly BC CA. If the explosion of the battleship Maine was within the ship, the plates must have been blown outward; if it was outside the ship, the plates must have been blown inward; then the converses of these statements are also true, and since it was found that the plates were bent in, the explosion must have been outside the ship. The truth of converse statements may also be shown by elimination of possibilities, that is by showing that all but one of the possibilities are untrue, and that the one remaining is therefore the true one because it is the only remaining possibility. For instance, in the foregoing geometrical illustration, given the same conditions, to find what is true if <A><B. If BC were equal to CA, then A would be equal to B, which is not true. If BC were < CA, then A would be <≤B, which is not true; therefore BC is >CA, because that is the only remaining possibility. Order and Arrangement of a Proof. In writing out a formal proof, it is best to have a definite order of arrangement that will show all the important details of the proof. The following arrangement is recommended for all propositions: STATEMENT OF PROPOSITION Diagram Given. The conditions of the proposition. To prove. The conclusion. (Both condition and conclusion should be in terms of the figure. If it is a construction, Required should take the place of To prove.) Proof. 1. The proof should follow in numbered steps. (If it is a construction, the proof may be divided into Construction and Proof.) ILLUSTRATION A line and a point outside determine a plane. Given. To prove. Line AB and external point P. AB and P determine a plane. Proof. 1. Let K and I be two points on AB, then K, L, and P determine a plane (definition of plane, § 5, p. 233). 2. But AB is entirely in this plane, for two of its points are in the plane (axiom of line and plane, § 6, p. 234). 3. Therefore AB and P determine a plane, for they lie in one, and but one, plane. SECTION III. THE REPRESENTATION OF SOLID GEOMETRY FIGURES Since in solid geometry three-dimensional figures are usually represented on the paper or on the blackboard, that is, on a surface of but two dimensions, the student needs some idea of how to make the figures appear solid. While there is neither space nor time for a study of drawing, it is believed that the following brief directions may be of assistance in giving this idea. Unless the student has given some time to the study of perspective, the parallel line method will be the simplest for him to use. In this, a plane, or rather, since a plane is unlimited in extent, -that portion of it shown in the figure, is, in general, represented by a parallelogram. It follows that many solids, since they are bounded by planes, can be drawn by the use of sets of parallel lines. This, of course, is not true of solids the conditions of which require that some portion of a plane shown in the figure shall be a circle, or a polygon other than a parallelogram. Where parallel lines are used, one set is usually drawn horizontally, the other somewhat inclined to the vertical. There are a few general rules that are of assistance in drawing all the figures, the most important being the following: The figures are usually supposed to be viewed from a point a little above, and either directly in front, or a little to one side. Lines nearer to the observer should be somewhat heavier than those at the back of the figure. As planes are supposed to be opaque, all lines apparently covered by them should be left out entirely, or, if needed in the figure, should be dotted. Where it seems desirable, lines that should not appear in the figure may be drawn and afterwards erased. A circle, unless viewed from a point directly opposite the center, appears elliptical, the ellipse being narrower as the eye is nearer its plane. It should be noted that two lines that intersect in a drawing do not necessarily intersect in the three-dimensional figure, and that the actual length of a line, or the size of an angle, may be very different from its representation, as each part is drawn so as to appear to be of the right size in the three-dimensional figure represented. The Use of Squared Paper. At the beginning, squared paper will be found very convenient for drawing the solid geometry figures, for with it equal lines, parallel lines, and perpendicular lines can be drawn with no instrument but the straightedge. As the student gains skill in drawing he should not confine himself to this method, but should draw accurate figures on unruled paper, using compasses and ruler. He should also draw reasonably accurate freehand figures. If models of the solid figures are used in addition to the drawings, the two together will give the best possible idea of the three-dimensional figures. The following explanation shows how to draw parallel or perpendicular lines on squared paper: (1) Parallel Lines. (Fig. 1). Draw the lines so that they connect points the same distance apart, measured along the lines of the paper, to the right or left, and up or down. For example, in Fig. 1, to draw a line from C parallel to AB, note that AB extends to the right 5 spaces, and up 15 spaces; therefore count from C to the right 5 spaces, and up 15 spaces to D, and draw CD. This also makes the lines equal, as can be seen by this illustration, where CD AB. If the lines are not to be of the same length, the same method is used, and that part of CD that is needed is taken. (2) Perpendicular Lines. (Fig. 2). Count as before, but draw the new line so that one direction is reversed, that is, up for down, left for right, etc., and so that the two numbers are interchanged. In Fig. 2, to draw a line from P perpendicular to AB, note that AB extends 10 spaces down, and 15 spaces to the right, so draw from P to a point o, 15 spaces down, and 10 spaces to the left, or, 15 spaces up, and 10 spaces to the right. The Plane (Figs. 3, 4). Draw a parallelogram, usually with two edges horizontal. The other positions in which a plane most frequently appears will be taken up in the following figures. If, for any reason, it is desirable to emphasize the fact that the plane is unlimited in extent, one pair of edges can be broken, as in Fig. 4. Two Lines in Space (Figs. 5, 8). Intersecting or parallel lines do not differ from those in plane geometry. Two lines not in the same plane, called non-coplanar lines, cannot be distinguished from intersecting lines, unless there is more in the figure to help show their relative position. In Fig. 5, the plane M appears to contain CD, and so makes it appear that AB and CD are not in the same plane. In Fig. 8, AB and RS are not in the same |