Dem. M+0=2 R. A. and M+X=2 R. A. (106.) Therefore M+OM+X; for it is an axiom that things which are equal to the same thing are equal to one another. MM, therefore O=X, for it is also an axiom, that if equals be taken from equals the remainders will be equal. In a similar manner it may be demonstrated that MN; and in general, that the vertical angles made by any number of straight lines crossing at one point are equal to each other. 108. Two external-internal angles are equal one to the other, (fig. 84.) A=E, B=F, &c. For two parallel lines have a similar position, one with the other; therefore a straight line crossing them has the same inclination to each, that is, it makes equal angles with each. 109. The two interior angles upon the same side taken together are equal to 2 R. A., (fig. 84.) D+F= 2 R. A.; C+E=2 R. A. Dem. B=F (108) and D=D. It is an axiom that if equals be added to equals the sums will be equal; therefore B+D=F+D. But BD2 R. A., (106); therefore F+D=2 R. A. 110. Two alternate-internal angles are equal one to the other, (fig. 84.) C=F, D= E, &c. Dem. C+E 2 R. A. (109), and FE 2 R. A. (106); therefore C+E=F+E; therefore C=F. 111. It may also be demonstrated that the alternateexternal angles are equal to each other, (for example A H; B=G;) that two alternate-internal-external angles taken together are equal to 2 R. A.; (that is, A+ = F=2 R. A., B+ E=2 R. A.;) and that two exterior angles upon the same side are together equal to 2 R. A. (that is, A+ G= 2 R. A.; B +H=2 R. A.) 112. Reciprocally, we may infer that two straight lines are parallel, if, when crossed by a third straight line, the following propositions are true; and the reverse of this, that two straight lines are not parallel, if these propositions are not true, viz., if, The external-internal angles are equal. The interior angles upon the same side are together equal to 2 R. A. The exterior angles upon the same side are together equal to 2 R. A. The alternate-external-internal angles are together equal to 2 R. A. 113. If two straight lines AB and CD are each parallel to a third line EF, they are parallel to each other (fig. 85.) Dem. Draw a straight line crossing the other three. Then, because AB is parallel to EF, therefore angle O =Y (108), and because CD is parallel to EF, therefore angle X=Y; therefore 0=X, consequently AB is parallel to CD. 114. Two straight lines are parallel to each other, if each is perpendicular to a third line; for the interior angles upon the same side taken together will be equal to 2 R. A. 5. MENSURATION OF ANGLES. 115. For the measure of angles upon paper we make use of a protractor. For measuring angles in the field various instruments are used, more or less complicated. The most simple is a graduated circle, (which may be made of metal, wood, or pasteboard,) with an index moving on a pivot in the middle. Place the instrument at the vertex of the angle to be measured, and make the index coincide in direction with one of the sides of the angle; then move the index until it coincides in direction with the other side of the angle, noting the number of degrees on the graduated circle which it passes over; this number will be the magnitude of the angle. For determining directions which shall be perpendicular to the surface of still water, that is, to a horizontal surface, a leaden weight hanging freely from a string, and called a plumb, is used. Positions which are parallel to the direction of the plumb, are perpendicular to the horizon, that is vertical. Masons use the plumb in building walls. For determining horizontal positions a plumb level, (fig. 88,) is frequently used. It consists of a wooden frame made in the form of an isosceles triangle, with a plumb attached to it. The base of the triangle is placed upon the surface the position of which is to be determined; if the plumb falls directly over the marked centre point of the base, then this base, and consequently the surface upon which it stands, is horizontal. This kind of level is sometimes made in the form of the letter L. The two parts must be exactly perpendicular, one to the other; a plumb is suspended froin the top of the vertical ruler; and if the string coincides exactly with the edge of this ruler, then the other ruler must be horizontal. This kind of level is also made in the form of an inverted T. viz., L. But the spirit level is the most accurate, and the one most commonly used. It consists of a cylindrical glass tube filled with spirits of wine, excepting a small portion containing air; the ends of the tube being hermetically sealed. The bubble of the air, being the lightest part of the contents of the tube, will always run towards that end which is highest; but when the tube is. horizontal it will have no tendency to either end. The bore of the tube is not exactly cylindrical, but it is slightly curved, the convex side being upward; therefore the bubble will rest in the middle of the tube, when the tube is horizontal. The tube is fitted into a block of wood, the bottom of which is exactly parallel to the tube; so that when the bottom of the block is horizontal, the bubble will be exactly between two scratches marked on the top of the tube to show the middle. IV. FIGURES. 1. FIGURES IN GENERAL. 116. We defined a line to be the path described by the motion of a point. In a similar manner we may say that a surface is the space described by a line moved in any direction but that of its length. For example, if you suppose a straight line to be turned on a pivot at one extremity or in the middle, it will describe a circle; if it be moved in a direction perpendicular to its length it will describe a parallelogram. If we suppose a curved line to revolve on an axis connecting its extremities it will describe a curved surface. A line has no thickness, consequently a surface has no thickness. A line has length, and its motion makes breadth, consequently a surface has extension in two directions, viz., length and breadth. 2. TRIANGLES IN GENERAL. 117. In every triangle one side is less than the sum of the other two. For a straight line is the shortest distance between two points; therefore (fig. 87) AB<AC +CB. 118. In every triangle the 3 angles are together equal to two right angles. Dem. Through the vertex of the angle A, (fig. 87,) draw a straight line parallel to the opposite side BC. Then O= B, and X=C (110). Consequently O+X +A-B+C+A. But 0+X+A= 2 R. A. (34); therefore B+C+A=2 R. A. 119. A corollary is a consequence which follows directly from a proposition. From the preceding proposition we have the following corollaries. 120. Cor. 1. In every triangle there can be but 1 R. A., and but 1 obtuse angle; there must be at least 2 acute angles. 121. Cor. 2. In every right triangle the 2 acute angles are together equal to 90°. In every obtuseangled triangle the 2 acute angles are together less than |