PROBLEM A. Describe an equilateral triangle upon a given straight line. Let AB be the given straight line. With A as centre, and radius AB, describe the circle BCD, and with B as centre and radius BA describe the circle ACE. Let these circles intersect in C; join CA, CB. Then ABC shall be the equilateral triangle required. .: A is the centre of the o BCD, . AB is = AC, and :: B is the centre of the O ACE, .. AB is -- BC, i AC is = BC; .. ABC is an equilateral triangle, and it has been described upon AB. Q. E. F. PROBLEM B. Let DAE be the given angle. With centre A describe a circle cutting AD, AE in the points B and C. With centres B and C describe equal circles cutting one another in F. Join AF; then 2 DAE shall be bisected by AF. Then : in the as ABF, ACF, BA is = CA, BF=CF, and AF is common ; .: the as ABF, ACF are equal in all respects, (1. 5) and : the BAF is = the 2 CAF, ... DAE has been bisected by AF. Q. E. F. PROBLEM C. Let AB be the given straight line. With centres A and B describe two circles having equal radii intersecting in C, D. Join CD cutting AB in E; then AB shall be bisected in E. spectively in all respecto (1.5) Then in the As ACD, BCD, i. AS ACD, BCD are equal in all respects, (1. 5) Hence in the As ACE, BCE, AC, CE and the included 2 ACE are respectively=BC, CE and the included - BCE; .. AS ACE, BCE are equal in all respects, and .. AE is = BE; (1.1) . .. AB has been bisected in E. , Q. E. F. RIGHT ANGLES. DEFINITION. When one straight line, standing on another straight line, makes the adjacent angles equal to one another, each of them is called a right angle; and the straight line which stands upon the other is said to be perpendicular to it. PROPOSITION VI. If two straight lines cut one another the opposite angles shall be equal. A Let two intersecting straight lines form the four angles A, B, H, K. Then shall the opposite angles A, B be = one another. For if the figure were taken up, reversed, and placed so that each of the arms of H might fall along the former position of the other arm; then each of these lines produced would fall along the former position of the other. Thus the arms of A would fall along the former positions of the arms of B; .. the angles A, B are equal to one another. Similarly the angles H, K may be proved equal. COR. If DEF be any angle, the adjacent 2 DEK formed by producing one of its arms FE is = the adjacent . FEL formed by producing the other arm. by producing its arms as the adjacent For they are opposite angles. .. if an angle = one of its adjacent angles it is also = the other, .. if one straight line is 1 to another, the latter is 1 to the former. Thus each of the arms of a right 2 is 1 to the other. |