22. Two straight lines which lie in the same plane and have different directions must meet if sufficiently prolonged; and must have one, and but one, point in common. Conversely Two straight lines lying in the same plane which do not meet have the same direction; for if they had different directions they would meet, which is contrary to the hypothesis that they do not meet. Two straight lines which meet have different directions; for if they had the same direction they would never meet (§ 21), which is contrary to the hypothesis that they do meet. ON PLANE ANGLES. 23. DEF. An Angle is the difference in direction of two lines. The point in which the lines (prolonged if necessary) meet is called the Vertex, and the lines are called the Sides of the angle. An angle is designated by placing a letter at its vertex, and one at each of its sides. In reading, we name the three letters, putting the letter at the vertex between the other two. When the point is the vertex of but one angle we usually name the letter at the vertex only; thus, in Fig. 1, we read the angle by calling it angle A. But in Fig. 2, H is the common vertex of two angles, so that if we were to say the angle H, it would not be known whether we meant the angle marked 3 or that marked 4. We avoid all ambiguity by reading the former as the angle E HD, and the latter as the angle EHF. The magnitude of an angle depends wholly upon the extent of opening of its sides, and not upon their length. Thus if the sides of the angle BA C, B namely, AB and AC, be prolonged, their extent of opening will not be altered, and the A size of the angle, consequently, will not be changed. 24. DEF. Adjacent Angles are angles. having a common vertex and a common side between them. Thus the angles CDE and CDF are adjacent angles. E 25. DEF. A Right Angle is an angle included between two straight lines which meet each other so that the two adjacent angles formed by producing one of the lines through the vertex are equal. Thus if the straight line AB meet the straight line CD so that the adjacent angles ABC and ABD are equal to one another, each of these angles is called a right angle. 26. DEF. Perpendicular Lines are lines which make a right angle with each other. 27. DEF. An Acute Angle is an angle less than a right angle; as the angle BA C. 28. DEF. An Obtuse Angle is an angle greater than a right angle; as the angle DEF. D E 29. DEF. Acute and obtuse angles, in distinction from right angles, are called oblique angles; and intersecting lines which are not perpendicular to each other are called oblique lines. 30. DEF. The Complement of an angle is the difference between a right angle and the given angle. Thus ABD is the complement of the angle DBC; also DBC is the complement of the angle A B D. A D B 31. DEF. The Supplement of an angle is the difference between two right angles and the given angle. Thus A CD is the supplement of the angle D C B; also D C B is the supplement of the angle AC D. 32. DEF. Vertical Angles are angles which have the same vertex, and their sides extending in opposite directions. Thus the angles AOD and COB are vertical angles, as also the angles AO C and DO B. ON ANGULAR MAGNITUDE. 33. Let the lines B B' and A A' be in the same plane, and let BB' be perpendicular to A A' at the point 0. Suppose the straight line OC to move in this plane from coincidence with O A, about the point 0 as a pivot, to the position O C; then the line OC describes or generates the angle A O C. A' The amount of rotation of the line, from the position 0 A to the position OC, is the Angular Magnitude A O C. If the rotating line move from the position OA to the position O B, perpendicular to O A, it generates a right angle; to the position O A' it generates two right angles; to the position O B', as indicated by the dotted line, it generates three right angles; and if it continue its rotation to the position O A, whence it started, it generates four right angles. Hence the whole angular magnitude about a point in a plane is equal to four right angles, and the angular magnitude about a point on one side of a straight line drawn through that point is equal to two right angles. 34. Now since the angular magnitude about the point is neither increased nor diminished by the number of lines which radiate from that point, the sum of all the angles about a point in a plane, as AOB+BOC+COD, etc., in Fig. 1, is equal to four right angles; and the sum of all the angles about a point on one side of a straight line drawn through that point, as AOB BOC+ COD, etc., Fig. 2, is equal to two right angles. Hence two adjacent angles, O CA and OC B, formed by two straight lines, of which one is produced from the point of meeting in both directions, are supplements of each other, and may Ā be called supplementary adjacent angles. ON THE METHOD OF SUPERPOSITION. H C B 35. The test of the equality of two geometrical magnitudes is that they coincide point for point. Thus, two straight lines are equal, if they can be so placed that the points at their extremities coincide. Two angles are equal, if they can be so placed that their vertices coincide in position and their sides in direction. In applying this test of equality, we assume that a line may be moved from one place to another without altering its length; that an angle may be taken up, turned over, and put down, without altering the difference in direction of its sides. B'C', so that the vertex B shall fall on B', then if the side BA fall on B'A', the angle ABC equals the angle A'B' C"; if the side BA fall between B'C' and B'A' in the direction B' D, the angle ABC is less than A'B' C"; but if the side B A fall in the direction B'E, the angle A B C is greater than A' B' C'. This method of superposition enables us to add magnitudes of the same kind. Thus, if we have two straight lines A B and CD, by A C А A BC D B placing the point Con B, and keeping C D in the same direction with AB, we shall have one continuous straight line AD equal to the sum of the lines A B 36. DEF. A Demonstration is a course of reasoning by which the truth or falsity of a particular statement is logically established. 37. DEF. A Theorem is a truth to be demonstrated. 38. DEF. A Construction is a graphical representation of a geometrical conception. 39. DEF. A Problem is a construction to be effected, or a question to be investigated. |