BOOK II. ON ANGLES AND PROPORTIONAL LINES. CHAPTER I. ON ANGLES. SECTION I. ON ANGLES GENERALLY. DEFINITIONS. "A plane angle is the inclination of two lines to one another in a plane, which meet together, but are not in the same direction." EUCLID. "When two straight lines meet, the greater or lesser quantity by which they recede from each other, as to their position, is called an angle." LEGENDRE. "An angle is the infinite portion of a plane contained between two straight lines which cut each other." BERTRAND (of Geneva). "The figure formed by two straight lines which cut each other is called an angle." BLANCHET. 1. An angle is the figure formed by two straight lines which have one extremity in common; the straight lines which form the angle are its sides, and their common extremity is the vertex of the angle, or the angular point. As the straight line is infinite in magnitude and only one extremity of the sides of an angle is determined, the other extremity is at an infinite distance from the vertex. An angle is distinguished by the letter at its vertex. If two or more angles have the same vertex, each angle is distinguished by a letter at some point of each side, together with the letter at the vertex, taking care to read the latter between the two others. 2. The angular magnitude is the difference in the direction of the sides of the angle. Thus the magnitude of an angle is wholly independent of the length of its sides, but increases or diminishes according to their directions; so that the sides of an angle may be shortened or lengthened, made equal or unequal, without altering the magnitude of the angle itself. From the preceding definitions, it follows, that two angles are equal when the difference in the directions of their sides is the same. It is obvious that two angles, the sides of which coincide, are equal in magnitude, since the difference in direction of their sides is one and the same; also, if an angle is applied upon its equal, so that the vertex and one of the sides coincide, the other sides coincide also, and then the two figures form only one angle, although the other extremities of the sides may not coincide. If two indefinite straight lines have one extremity in common, one of them may be conceived to turn on their common point, and have at first a direction very little different from that of the other, so that the angle formed by these lines is very little, if reckoned from one side of one of the two lines, but is much greater if reckoned from the other side of the same line. Thus every angle has two magnitudes, one of which increases when the other diminishes, and diminishes when the other increases.1 3. Each of the two magnitudes of an angle is the explement of the other. Two angles are explementary when the magnitude of one is equal to the explement of the other; each of the two angles is also the explement of the other. An angle contained in a plane divides it into two portions, one of which increases or diminishes with the angle, whilst the other increases or diminishes with the explement of the angle. 4. The portion of the plane containing an angle, which 'Throughout The Elements, when the magnitude of an angle is spoken of without anything further being said, the lesser magnitude only is meant. increases or diminishes as the angle increases or diminishes, is the angular space. All points of the angular space are inside the angle, or in the opening of the angle; all points of the plane not contained in the angular space, are outside the angle. The two sides of an angle may be oblique or perpendicular to one another; they may even be the continuation of each other; that is, have opposite directions, so that they form a straight line: in the latter case the angle is called a straight angle. Thus 5. A straight angle is an angle each side of which is the continuation of the other. An angle which is less than a straight angle is a projecting angle; an angle which is greater than a straight angle is a receding angle. Hence, the two sides of a straight angle form a straight line. 6. Two angles which have the same vertex and one common side are adjacent to each other, or simply adjacent. If one side of an angle be produced beyond the vertex, the two angles thus formed, having one side in common, will be adjacent to each other. If both sides of an angle be produced beyond the vertex, each of the four angles thus formed with a common vertex will be adjacent to two others; the opposite angles, that is those that are not adjacent, are said to be the vertical of each other. Thus 7. Vertical angles are two angles such that the sides of the one are the continuation of the sides of the other beyond the vertex. 8. The line which joins two points of the sides of an angle subtends the angle. 9. Two points of the sides of an angle, which are equally distant from the vertex, are symmetric points of the angle. |