THEOREM III. 47. CONVERSELY.-If the sum of two adjacent angles equals two right angles, their exterior sides lie in the same straight line. To prove that AC and BC lie in the same straight line. Draw EC. If EC and BC lie in the same straight line, .. AC and BC lie in the same straight line. Q. E. D. THEOREM IV. 48. If two straight lines intersect, the opposite or vertical angles are equal. Likewise we can prove that c = L d. Q. E. D. 49. COR. 1.-If two straight lines intersect, the sum of the four angles formed equals four right angles. 50. COR. 2.—The sum of all the angles that can be formed at a common point equals four right angles. THEOREM V. 51. From a point without a straight line, only one perpendicular can be drawn to that line. To prove that only one can be drawn from P to AB. Draw the oblique line PC. revolve PC so as to decrease With the point P fixed, a and increase b, while the common vertex moves in the direction CA. At some position of the line, as PD, the adjacent angles are equal. Then PD is L to AB. (20) There is only one position of the line in which the angles are equal. ... only one can be drawn from P to AB. Q. E. D. THEOREM VI. 52. From a point without a straight line, a perpendicular is the shortest distance to that line. Let AB be a straight line, C any point without it, CE a L, and CF any oblique line. On AB as an axis, revolve the plane of CEF till it falls in the plane of DEF. Since sa and b are Ls, the line CE takes the direction ED, the point C falling on D. THEOREM VII. 53. Any point in the perpendicular erected at the middle point of a straight line is equally distant from the extremities of that line. Let P be any point in CD which is to AB at its middle point D, and let AP and BP be drawn. On CD as an axis, revolve APD till it falls in the plane 54. COR. 1.-If a point is equally distant from the extremities of a straight line, it lies in the perpendicular erected at the middle point of that line. 55. COR. 2.-If each of two points in a straight line is equally distant from the extremities of another straight line, the former is perpendicular to the latter at its middle point. |