= 1. The triangle BCF is equal to the triangle BCA; for the right angle CBF = CBA, the side CB is common, and the side BF BA; therefore the third side CF is equal to the third side AC (36). But AF <AC+ CF (40), and AB half of AF is less than AC half of AC+CF, that is, the perpendicular is less than any one of the oblique lines. 2. If BE = BC, then, as AB is common to the two triangles ABE, ABC, and the right angle ABE = ABC, the triangle ABE is equal to the triangle ABC, and AE = AC. 3. In the triangle DFA, the sum of the sides AD, DF, is greater than the sum of the sides AC, CF (41); therefore AD half of AD+DF is greater than AC half of AC + CF, and the oblique line, which is more remote from the perpendicular, is greater than that which is nearer. 53. Corollary 1. The perpendicular measures the distance of any point from a straight line. 54. Corollary II. From the same point, there cannot be drawn three equal straight lines terminating in a given straight line; for, if this could be done, there would be on the same side of the perpendicular two equal oblique lines, which is impossible. Fig. 32 THEOREM. 55. If from the point C (fig. 32), the middle of the straight line AB, a perpendicular EF be drawn; 1. each point in the perpendicular EF is equally distant from the two extremities of the line AB; 2. any point without the perpendicular is at unequal distances from the same extremities A and B. Demonstration. 1. Since AC CB, the two oblique lines AD, DB, are drawn to points which are at the same distance from the perpendicular. They are therefore equal (52). The same reasoning will apply to the two oblique lines AE, EB, also to AF, FB, &c. Whence each point in the perpendicular EF is equally distant from the extremities of the line AB. one 2. Let I be a point out of the perpendicular; join IA, IB, of these lines must cut the perpendicular in D; join DB, then DB DA. But the line IB <ID + DB and = therefore IB <IA; that is, any point without the perpendicular is at unequal distances from the extremities of AB. THEOREM. 56. Two right-angled triangles are equal, when the hypothenuse and a side of the one are equal to the hypothenuse and a side of the other, each to each. Demonstration. Let the hypothenuse AC DF (fig. 33), and Fig. 33. the side AB = DE; the right-angled triangle ABC will be equal to the right-angled triangle DEF. The proposition will evidently be true, if the third side BC be equal to the third side EF. If it be possible, let these sides be unequal, and let BC be the greater. Take BG EF, and join AG; then the triangle ABG is equal to the triangle DEF, for the right angle B is equal to the right angle E, the side AB= DE and the side BG EF; therefore these two triangles being equal (36), AG = DF; and, by hypothesis, DF AC; whence AG AC. But AG cannot be equal to AC (52); therefore it is impossible that BC should be unequal to EF, that is, it is equal to it, and the triangle ABC is equal to the triangle DEF. THEOREM. 57. If two straight lines, AC, BD, (fig. 35), are perpendicular to Fig. 35. a third AB, these two lines are parallel, that is, they will not meet, however far they are produced (12). Demonstration. If they could meet in a point O on one side or the other of the line AB, there would be two perpendiculars drawn from the same point O to the same straight line AB, which is impossible (50). LEMMA, 58. The straight line BD (fig. 35), being perpendicular to AB. Fig. 35. if another straight line AE make with AB an acute angle BAE, the straight lines BD, AE, being produced sufficiently far, will meet. Demonstration. From any point F, taken in the direction AE, let fall upon AB the perpendicular FG; the point G will not fall upon A, since the angle FAB is less than a right angle; still less can it fall upon H in BA produced, for then there would be two perpendiculars KA, KH, let fall from the same point K upon the same straight line AH. The point G then must fall, as the figure represents it, in the direction AB. Let there be taken in the line AE another point L at a distance AL greater than AF, and let there be drawn to AB the perpendicular LM; it may be shown, as in the preceding case, that the point M can neither fall upon G nor upon any point in the direction GA, so that the distance AM will be necessarily greater than AG. It may be observed, moreover, that if the figure is constructed with care, and AL be taken double of AF, we shall find that AM is exactly double of AG; also, if AL be taken triple of AF, we shall find that AG is triple of AG, and in general there will always be the same ratio between AM and AG, that there is between AL and AF. From this proportion it follows, not only that the right line AE, must meet BD, if the two lines are produced sufficiently far, but also that we may even assign upon AE the distance of the point of meeting of the two lines. This distance will be the fourth term of the proportion, AG: AB:: AF: x. 59. Scholium. The foregoing explanation, founded upon a relation which is not deduced from reasoning merely, and for which recourse is had to measures taken upon a figure accurately constructed, has not the same degree of strictness, as the other demonstrations of elementary geometry. It is given here only as a simple method, by which one may satisfy himself of the truth of the proposition. We shall resume the subject with a view to a rigorous demonstration in the third of the notes subjoined to these elements. Fig. 36. THEOREM. 60. If two straight lines AC, BD, (fig. 36), make with a third AB two interior angles CAB, ABD, the sum of which is equal to two right angles, the two lines AC, BD, are parallel. Demonstration. From the point G, the middle of AB, draw the straight line EGF perpendicular to AC; this line will be perpendicular to BD. Indeed the sum GAE÷GBD is, by hypothesis, equal to two right angles, and the sum GBF+GBD is also equal to two right angles (28); taking therefore from each GBD we shall have the angle GAE = GBF. Besides, the angles AGE, BGF, are equal, being vertical angles; therefore the triangles AGE, BGF, have a side and the two adjacent angles of the one res pectively equal to a side and the two adjacent angles of the other; they are therefore equal (38), and the angle BFG = AEG; but the angle AEG is, by construction, a right angle, conse quently the straight lines AC, BD, are perpendicular to the same straight line EF, therefore they are parallel (57). THEOREM. 61. If two straight lines, AI, BD (fig. 36), make with a third Fig. 36. line AB two interior angles BAI, ABD, the sum of which is less than two right angles, the lines AI, BD, produced, will meet. Demonstration. Draw AC making the angle CAB = ABF, that is, in such a manner as to make the two angles CAB, ABD, together equal to two right angles, and finish the construction, as in the preceding theorem. Since the angle AEK is a right angle, AE is a perpendicular, and consequently less than the oblique line, AK; whence in the triangle AEK the angle AKE, opposite to the side AE, is less than the right angle AEK, opposite to the side AK (49). Therefore the angle IKF, equal to the angle AKE, is less than a right angle, and the lines KI, FD, being produced, must meet (58). 62. Scholium. If the lines AM and BD make with AB two angles BAM, ABD, the sum of which is greater than two right angles, then the two lines AM, BD, would not meet above AB, but they would meet below it. For the two angles BAM, BAN, would together be equal to two right angles, as also the two angles, ABD, ABF, and the four angles taken together would be equal to four right angles. But the sum of the two angles BAM, ABD, is greater than two right angles, therefore the sum of the two remaining ones BAN, ABF, is less than two right angles, and the two right lines AN, BF, being produced, must meet. 63. Corollary. Through a given point A only one line can be drawn parallel to a given line BD. For there can be only one line AC, which makes the sum of the two angles BAC, ABD, equal to two right angles; this is the parallel required, every other line Al or AM would make the sum of the interior angles less or greater than two right angles, therefore it would meet the line BD. Fig. 37. THEOREM. 64. If two parallel straight lines AB, CD (fig. 37), meet a third line EF, the sum of the interior angles upon the same side AGH, GHC, will be equal to two right angles. Demonstration. If this sum were greater or less than two right angles, the two straight lines AB, CD, would meet on one side or the other of EF, and would not be parallel (61). 65. Corollary 1. If GHC be a right angle, AGH will also be a right angle; therefore every line, which is perpendicular to one of the parallels, is also perpendicular to the other. 66. Corollary 11. Since the sum AGH+GHC is equal to two right angles, and the sum GHD+ GHC is also equal to two right angles, if we take away the common part GHC, we shall have the angle AGH GHD. Besides, AGH = BGE, and = GHD= CHF (34); therefore the four acute angles AGH, BGE, GHD, CHF, are equal to each other; the same may be proved with respect to the four obtuse angles AGE, BGH, GHC, DHF. It may be observed, moreover, that by adding one of the four acute angles to one of the four obtuse angles, the sum will always be equal to two right angles. 67. Scholium. The angles of which we have been speaking, compared, two and two, take different names. We have already called the angles AGH, GHC, interior upon the same side; the angles BGH, GHD, have the same name; the angles AGH, GHD, are called alternate-internal, or simply alternate; the same may be said of the angles BGH, GHC. Lastly, we denominate internal-external the angles EGB, GHD, and EGA, GHC, and alternate-external EGB, CHF, and AGE, DHF. This being premised, we may regard the following propositions as already demonstrated. 1. The two interior angles upon the same side, taken together, are equal to two right angles. 2. The alternate-internal angles are equal, as also the internalexternal, and the alternate-external. Reciprocally, if in this second case, two angles of the same name are equal, we may infer that the lines to which they are referred are parallel. Let there be, for example, the angle AGH= GHD; since GHC + GHD is equal to two right angles, |