GEOMETRICAL AXIOMS. 1. Magnitudes that can be made to coincide are equal. 2. Through two points there can be made to pass one, and only one, straight line: and this may be indefinitely prolonged either way. Hence, a. Any straight line may be made to fall on any other straight line with any given point on the one on any given point on the other; B. Two straight lines which meet in one point cannot meet again unless they coincide. SECTION I. ANGLES AT A POINT. [An angle is a simple concept incapable of definition, properly so-called, but the nature of the concept may be explained as follows, and for convenience of reference the explanation may be reckoned among the definitions.] DEF. 7. When two straight lines are drawn from the same point, they are said to contain, or to make with each other, a plane angle. The point is called the vertex, and the straight lines are called the arms, of the angle. A line drawn from the vertex and turning about the vertex in the plane of the angle from the position of coincidence with one arm to that of coincidence with the other is said to turn through the angle: and the angle is greater as the quantity of turning is greater. Since the line may turn from the one position to the other in either of two ways, two angles are formed by two straight lines drawn from a point. These angles (which have a common vertex and common arms) are said to be conjugate. The greater of the two is called the major conjugate, and the smaller the minor conjugate, angle. When the angle contained by two lines is spoken of without qualification, the minor conjugate angle is to be understood. It is seldom requisite to consider major conjugate angles before Book III. When the arms of an angle are in the same straight line, the conjugate angles are equal, and each is then said to be a straight angle. DEF. 8. When three straight lines are drawn from a point, if one of them be regarded as lying between the other two, the angles which this one (the mean) makes with the other two (the extremes) are said to be adjacent angles: and the angle between the extremes, through which a line would turn in passing from one extreme through the mean to the other extreme, is the sum of the two adjacent angles. DEF. 9. The bisector of an angle is the straight line that divides it into two equal angles. DEF. 10. When one straight line stands upon another straight line and makes the adjacent angles equal, each of the angles is called a right angle. OBS. Hence a straight angle is equal to two right angles; or, a right angle is half a straight angle. DEF. II. A perpendicular to a straight line is a straight line that makes a right angle with it. C DEF. 12. An acute angle is that which is less than a right angle. DEF. 13. An obtuse angle is that which is greater than one right angle, but less than two right angles. DEF. 14. A reflex angle is a term sometimes used for a major conjugate angle. DEF. 15. When the sum of two angles is a right angle, each is called the complement of the other, or is said to be complementary to the other. DEF. 16. When the sum of two angles is two right angles, each is called the supplement of the other, or is said to be supplementary to the other. DEF. 17. The opposite angles made by two straight lines that intersect are called vertically opposite angles. THEOR. 1. All right angles are equal to one another. Let ABC be a right angle formed by the straight line AB standing on the straight line CBD, EFG a right angle formed by the straight line EF standing on the straight line GFH : then shall the angle ABC be equal to the angle EFG. H Apply the straight line CBD to the straight line GFH, so that the point B may fall on the point F, Ax. 2a. and the straight line BA on the same side of GFH as FE; the line BA shall fall on the line FE. For if it falls otherwise it must fall either within the angle EFH or the angle EFG. Let it fall within the angle EFH, as FK. Then, because the angle KFG is a right angle, therefore it is equal to the angle KFH; but the angle EFH is greater than the angle KFH, Def. 10. Ax. a. therefore the angle EFH is also greater than the angle KFG, much more then is the angle EFH greater than the angle EFG. Ax. a. But because the angle EFG is a right angle, therefore it is equal to the angle EFH; Def. 10. therefore the angle EFG is both greater than, and equal to, the angle EFH, which is impossible, therefore the line BA does not fall within the angle EFH. In the same way it may be proved that the line BA does not fall within the angle EFG. Hence the line BA does fall on the line FE. Therefore the angle ABC coincides with the angle EFG, and therefore the angle ABC is equal to the angle EFG. Ax. 1. Q.E.D. COR. 1. At a given point in a given straight line only one perpen dicular can be drawn to that line. COR. 2. The complements of equal angles are equal. THEOR. 2. If a straight line stands upon another straight line, it makes the adjacent angles together equal to two right angles. Let the straight line AB stand upon the straight line CD : then shall the angles ABC, ABD be together equal to two right angles. If the angle ABC is equal to the angle ABD, each of them is a right angle, and therefore they are together equal to two right angles. But if the angle ABC is not equal to the angle ABD, Def. 10. let BE be a straight line standing upon the straight line CBD, so as to make the angle CBE equal to the angle EBD, then the angles CBE, EBD are two right angles. Def. 10. Now the angle ABC is equal to the two angles CBE, EBA, Ax. b. to each of these equals add the angle ABD, then the two angles ABC, ABD are together equal to the three angles CBE, EBA, ABD. Ax. d. Again, the angle EBD is equal to the two angles EBA, ABD, to each of these equals add the angle EBC, then the two angles EBD, EBC are together equal to the three angles CBE, EBA, ABD; Ax. d. |