BOOK i. PROP, xxiv. xxvm.] MATHEMATICS. PLANE GEOMETRY. 



547 



PROPOSITION XXIV. THEOREM. 



I/ two triangles (A B C, D E F) hare two sides (A B, A C) 

 of the one equal to two sides (D E, D F) of the other, 

 each to each, but the angle (BAG) contained by the tivo 

 tides of one of them greater than the angle (E D F) con- 

 tained by tlie two sides equal to them of the other ; the 

 base (B C) of that which has the greater angle shall be 

 greater than the base (E F) of the other. 

 Of the two aides D E, D F, let D E be the side which 



is not greater than the 



other. Make the angle 

 Pr.23. EDG =A, 



and make I) G = A C or 



+ Pr. 3. D F.f Draw 

 E G, G F, and let H be 

 the point where E G is 

 cut, either by D F or 

 by D F prolonged. 



Because D E is not 

 greater than D G, the angle D G E is not greater than 



. Pr. is. D E G ;* but D H G w greater than D E G, 

 . . D H G is greater than D G H, and . . D G is greater 



* Pr. 19. than D H ;t but D F = D G, . . D F is 

 greater than D H, . . H G F i apart ofDGF. Again : 

 the two sides B A, A C are = the two ED, D G, each to 



Hyp. ft Conzt. each, and the angle A= BUG,* .. 

 B C = E G. And because D F = D G, the angle D F G 



+ Pr. S. = D G F :f but it was proved that D G F is 



greater than EOF, .-. D F G is greater than E G F ; 



much more then is E F G greater than E G F, . . in the 



: F E G the angle E F G is greater than E G F, 



Pr. 19. . '. E G is greater than E F ;* but, as already 

 proved, E G = B C, . -. B C isgreater than E F. There- 

 fore, if two triangles, <tc. Q.E.D. 



PROPOSITION XXV. THEOREM. 

 If two triangles (A B C, D E F) have two tides (A B, A C) 

 of the one equal to two sides (1) E, D K) of the other 

 each to each, but the base (B C) of one greater than the 

 base (El-) of the other, the angle (A) contained by the 

 tides of that tcitA the greater base shatt be greater than 

 the angle (D) contained by the rides equal to those of 

 the other. 



For if A be not greater 

 than I), it must be either 

 equal to it or less. A is not 

 equal to D, for then B C 

 would be=EF; . Pr . 4 . 

 but it is not. A is not less 

 than 1), for then B C would 

 be less than E F :* Pr. 14. 

 but it is not.f As + Hyp. 

 therefore it is neither equal 

 to nor less than 1), A must 

 be greater than D, . . if two triangles, <tc. Q.E.D. 



PROPOSITION XX VL THEOREM. 

 If two triangles (ABC, DBF) have two angles (B, C) of 

 the one equal to two (E, F) of the other, each to each ; and 

 one tide equal to one side etz. , either the sides (B C. E F) 

 adjacent to the equal angles, or the sides (A B, 'D E) 

 opposite to equal angles in each ; then shall the other 

 tides be equal, each to each, ai-J, also the third angle of 

 the one to the third angle of the other. 

 First let B C E F, the sides adjacent to the angles 



that are equal each to 

 each. Then if A B, 

 D E be unequal, one 

 of them must be the 

 greater. Let A B be 

 the greater, and make 

 B G = D E,* . Pr . ,. 

 and draw G C : then 

 because BG=E D, and 



B C = E F,f + Hyp. 



and that the angle B = E, . '. the angle G C B = 

 Pr. 4. F:* hut the angle ACB = F,f .'. the angle 



* Hyp. U C B * A C B, the less to the greater ; which 



Ji 



H 



is impossible ; .'. A B is not unequal to D E, that is, it is 

 equal to it, . '. in the two triangles ABC, D E F, the 

 two sides A B, B C and the included angle B in the one, 

 are respectively equal to the two sides 1) E, E F and the 

 included angle E in the other, .-. AC=DF, and the 



Pr. 4. angle A= D. * 



Next let A B=D E, the sides opposite to equal angles ; 



in this case, likewise, 

 the other sides shall be 

 equal namely, A C = 

 D F, and B C = E F 

 and also the angle BAG 

 = 1). For, ifBC, EF 

 be unequal, let B C be 

 the greater, and make 

 B H = E F.' . p r . 3. 

 Join A, H : then be- 

 cause BH=EF, and AB=DE, the two A B, B H = 

 D E, E F, each to each ; and they contain equal angles ; 



Pr 4 .'. AH = DF,* and the angle BHA = EF1>: 

 + Hyp. but EF]> = BCA,t .'. BHA=BCA, the 



exterior angle equal to the interior and opposite, which 



Pr. 16. is impossible ; .'. B G is not unequal to E F, 

 .-. BC=EF: and AB = DE, .-. the two A B, B C= 

 D E, E F, each to each ; and they contain equal angles, 

 .'. AC=DF, and B AC=EDF, .-. if two triangles, 

 <fec. Q.E.D. 



PROPOSITION XXVII. THEOREM. 



If a straight line (E F) falling upon two other straight 

 lines ( A B, C D) makes the alternate angles ( A E F, E F D) 

 equal, these two straight lines shall be parallel. 

 For if they be / _ 



not parallel they A 



will meet, when 



prolonged, either - * 



towards B, D, or / D 



towards A, C. Let / 



them be prolonged and meet in the point G ; then G E F 



will be a triangle, and its exterior angle A E F must be 



Pr. 18. greater than the interior and opposite angle 

 + Hyp. E F G ;* but it is also equal to it,f which is 



impossible ; . . A B, C D, when prolonged, do not meet 

 towards B, D. In like manner it may be proved that 

 they do not meet towards A, C. . . they are parallel, . ' . 



rwf. . if a straight line, <fee. Q. E. D. 



PROPOSITION XXVIII. THEOREM. 



If a straight line (E F) falling upon two other straight 

 lines (A B, C D) make the exterior angle (E G B) equal 

 to theinterior and opposite angle (G H D) upon the same 

 side of the line ; or make the interior angles (B G H, 

 G H D) upon the same side together equal to two right 

 angles ; the two straight lines (A B, C D) shall be 

 parallel. 

 Because the angle EGB = the angle GHD,* and 



Hrp. EGB = AGH,t.'. AGH= GHD; and 

 + Pr. li. these are al- 



ternate angles, . . A B p. 



w parallel to CD.* \ 



Pr. 27. Again : be- _ * ' 

 cause the angles B G H, A. 



GHD are together = 

 two right angles,* and 



. Hvp. that A G H, C 

 B G H are together also 

 two right Angles, t . ' . 



\G 



\ 



I 



pr.u. AGH + BGH= BGH + GHD. Take 

 away BGH, then A G H = G H D ; and these are 



Pr. 27. alternate angles, . . A B is parallel to C D,* . . 

 if a straight line, <fec. Q.E.D. 



NOTE. Propositions XXVII. and XXVIII. clearly 

 prove the existence of parallel lines, or of lines such that, 

 however far they be prolonged, they can never meet. 

 And by aid of the first of these propositions, if a straight 

 line be given, one parallel to it may always be drawn 

 (see Prop. XXXI.) But, to proceed further in the doc- 

 trine of parallel lines, requires assent to a principle 



