1 SECTION I. OF STRAIGHT LINES AND ANGLES. QUERY I. In how many points can two straight lines cut each other? common; that is, could not a part of the line ED bend over and touch the line AB in M ? A. No. Q. Why not? A. Because there would be two straight lines, drawn between the same points E and M, which is impossible. (Truth 8.) QUERY II. If two lines have any part common, what must necessarily follow? A. They must coincide with each other throughout, and make but one and the same straight line. Q. How can you C prove this, for instance, of the two lines CA, MB, which have the part MA common? A. Because, between the two points A and M they cannot vary; otherwise there would be more than one straight line drawn between the two points M and A. Q. But is it not possible for either of the parts MC or AB to vary from the direction AM? A. No. Q. Why not? A. Because MA forming part of the line AB as well as of the line AC; MC and AB are but the continuations of the same straight line AM. Q. Will you now repeat the whole of your reasoning, and prove that two straight lines cannot have any part common, without coinciding with each other throughout, and making but one and the same straight line? If a straight line meets another, how great will be the sum of the two adjacent angles, which it makes with it, taking a right angle for the measure? A. It will be equal to two right angles. Q. How do you prove this of the two angles, formed by the line ED, meeting the line AC in the point D? A. Because, if in D A. M. D E you erect the perpendicular DM, the two angles ADE and CDE will occupy exactly the same space as the two right angles, ADM and CDM, formed by the meeting of the perpendicular; namely, all the space on one side of the line AC. Q. Can you prove the same of the sum of the two adjacent angles, formed by the meeting of any other two straight lines? on the same side of the straight line AC, taking again a right angle for the measure? A. It will also be equal to two right angles. Q. Why? A. Because, by erecting in that point B a perpendicular to AC, all these angles will be found to occupy the same space as the two right angles, made by the perpendicular. QUERY V. When two straight lines, AB, CD, cut each other, what relation will the angles which are opposite to each other at the vertex M, bear to each other? A. They will be equal to each other. A A. Because, if you add the same angle a, first to b, and then to e, the sum will, in both cases, be the same (equal to two right angles); and if the angle b were not equal to the angle e, this could not be ; and in the same manner I can prove that the two angles a and d are equal. Q. If the lines CD, AB, are perpendicular to each other, what remark can you make in relation to the angles d, cb, e, a? A. That each of these an gles will be a right angle. Q, And what is the sum of all the angles, a, b, c, d, e, f, around the same point, equal to ? A. To four right an- 0 gles. Q. Why? A. Because, if through B ab -D e α A M C a d f 24 P |