4. A line may be bisected; that is, divided into two equal parts. 5. An angle may be bisected. 6. A perpendicular may be drawn to a given line, either from a point without, or from a point on the line. 7. A line may be drawn, making with a given line an angle equal to a given angle. 8. A line may be drawn through a given point, parallel to a given line. NOTE. In making references, the following abbreviations are employed, viz. : A. for Axiom; B. for Book; C. for Corollary; D. for Definition; I. for Introduction; P. for Proposition; P. for Proposition; Prob. for Problem; Post. for Postulate; and S. for Scholium. In referring to the same Book, the number of the Book is not given; in referring to any other Book, the number of the Book is given. PROPOSITION I. THEOREM. If a straight line meet another straight line, the sum of the adjacent angles will be equal to two right angles. Let DC meet AB at C: then will the sum of the angles DCA and DCB be equal to two right angles. At C, let CE be drawn perpendicular to AB (Post. 6); then, by definition (D. 12), the angles A E D B ECA and ECB will both be right angles, and conse quently, their sum will be equal to two right angles. The angle DCA is equal to the sum ECA and ECD (A. 9); hence, of the angles ECA + ECD + DCB ; DCA + DCB But, ECD + DCB is equal to ECB (A. 9); hence, DCA + DCB = ECA + ECB. The sum of the angles ECA and ECB, is equal to two right angles; consequently, its equal, that is, the sum of the angles DCA and DCB, must also be equal to two right angles; which was to be proved. Cor. 1. If one of the angles DCA, DCB, is a right angle, the other must also be a right angle. Cor. 2. The sum of the angles BAC, CAD, DAE, EAF, formed about a given point on the same side of a straight line BF, is equal to two right angles. For, their sum is equal to D C E B F A the sum of the angles EAB and EAF; which, from the proposition just demonstrated, is equal to two right angles. DEFINITIONS. If two straight lines intersect each other, they form four angles about the point of intersection, which have received different names, with respect to each other. 2o. OPPOSITE, or VERTICAL ANGLES, are those which lie on opposite sides of both lines; thus, ACE and DCB, or ACD and ECB, are opposite angles. From the pro position just demonstrated, the sum of any two adjacent angles is equal to two right angles. PROPOSITION II. THEOREM. If two straight lines intersect each other, the opposite or vertical angles will be equal. Let AB and DE intersect at C: then will the opposite or vertical angles be equal. The sum of the adjacent angles ACE and ACD, is equal to two right angles (P. I.): the sum A E D B of the adjacent angles ACE and ECB, is also equal to two right angles. But things which are equal to the same thing, are equal to each other (A. 1); hence, ACD + ACE = ACD + DCB ; and, taking away the common angle ACD, we have, ACE DCB. Hence, the proposition is proved. Cor. 1. If one of the angles about C all of the others will be right angles also. is a right angle, For, (P. I., C. 1), D A E Cor. 2. If one line DE, is perpendicular to another AB, then will the second line AB be perpendicular to the first DE For, the angles DCA and DCB are right angles, by definition (D. 12); and from what has just been proved, the angles ACE and BCE are also right angles. Hence, the two lines are mutually perpendicular to each other. Cor. 3. The sum of all the angles ACB, BCD, DCE, ECF, FCA, that can be formed about a point, is equal to four right angles. B A F E ·D For, if two lines be drawn through the point, mutually perpendicular to each other, the sum of the angles which they form will be equal to four right angles, and it will also be equal to the sum of the given angles (A. 9). Hence, the sum of the given angles is equal to four right angles. PROPOSITION III. THEOREM. If two straight lines have two points in common, they will coincide throughout their whole extent, and form one and the same line. Let A and B be two points common to two lines: then will A the lines coincide throughout. Between A and B they must E B C D coincide (A. 11). Suppose, now, that they begin to separate at some point C, beyond AB, the one becoming ACE, and the other ACD. If the lines do separate at C, one or the other must change direction at this point; but this is contrary to the definition of a straight line (D. 4): hence, the supposition that they separate at any point is absurd. They must, therefore, coincide throughout; which was to be proved. Cor. Two straight lines can intersect in only one point. NOTE. The method of demonstration employed above, is called the reductio ad absurdum. It consists in assuming an hypothesis which is the contradictory of the proposition to be proved, and then continuing the reasoning until the assumed hypothesis is shown to be false. Its contradictory is thus proved to be true. This method of demonstration is often used in Geometry. |