I.COK I. PROP, xxxin. XL.] MATHEMATICS. PLANE GEOMETRY. 



649 



angles, are = twice as many right angles as ihe figure has 

 tides. 



COR. 2. All the exterior angles of any rectilineal figure 

 are together = four right angles. 



Because every interior angle 

 A B C, with its adjacent exte- 

 rior angle A B D, is = two 



Pr. 13. right angles,* . . all 

 the interior -+ all the exterior 

 are = twice as many right an- 



fles as there are sides to the 

 gure ; that is, by the above 

 corollary, to all the interior 

 angles -f- four right angles, .. 

 all the exterior angles are = four right angles. 



This rental kable property can scarcely fail to arrest the student's 

 special attention ; as, prt\ iou-ly to its demonstration, he would be 

 little likely to expect that if the sides of a rectilineal tiprure be pro- 

 longed, one after another, the exterior angles ttius formed would 

 have the same amount whether the figure had three sides, or three 

 thousand. 



PROPOSITION XXXIII. THEOREM. 



The straight lines (A C, B D) which join the extremities of 

 txo equal and parallel straight lines (A B, C D) towards 

 the tame parts, are also themselves equal and parallel. 

 Draw B C, which joins the extremities of the parallels 



towards opposite parts ; then the alternate angles A B C, 



Pr. 29. BCD are equal.* And because A B = C D, 

 and B C common to the 



two triangles A B C, A n 



DOB, the two sides A B, 

 B C and the included an- 

 gle are respectively = the 

 two D C, C B and the 

 included angle ; . '. A C = 

 t Pr. 4. BD,t and the angle ACB = DEC ; and these 



IT. 17. are alternate angles, . '. AC i* parallel to B D ;* 

 an I it was shown that AC = BD ; .-.the straight lines, 

 4c. Q. E. D. 



PROPOSITION XXXIV'.-TuBORFM. 

 The opposite sides and angles of a parallelogram A C D B 

 are equal, and the diagonal (B C) bisects it ; that is, 

 divides it into two equal parts. 



Dir A parallelogram i" a four-sided figure, of which the opposite 

 side* are parallel ; and the diagunal U the straight line joining two 

 of its opposite vertices. 

 Because A B is parallel 

 to CD, and BC meets 

 them, the angle A B C = 



rr.39. DCB;* and 

 because A C is parallel to 

 B 1 >, and B C meets them, 



t Pr. 29. the angle A C B = D B C,t . ' . the two tri- 

 angles ABC, 1) C B have two angles ABC, AC B, in 

 the one = I) C B, I) B C, in the other, each to each, and 

 the side B C, adjacent to the equal angles, common to 

 the two triangles ; .'. A B = C 1), and A C=B 1) ; and the 



Pr 26 angle A = 1).* Again : because the angle 

 A B C = I) C B, and the angle D B C = A C H, .-. the 

 whole angle. A B D = the whole angle A C D ; and it was 

 proved that A = D, . . the opposite sides and angles of a 

 parallelogram are equal. Also the diagonal bisects it : 

 for it has been shown that the triangle A C B has two 

 ides, and the included angle A = respectively to two 

 tides, and the included angle D in the triangle D B C, 



. p r 4. these triangles are equal,* . . the diagonal B C 

 liciJes the parallelogram into two equal parts. 



PROPOSITION XXXV. THEOREM. 

 "arallelograms (A BCD, EBCF) upon the same base 

 (B C) and between the same parallels (A F, B C) are 

 equal. 



Suppose, first, that the sides 

 _- A D, E V, opposite to the base 

 B C, terminate in the same point 

 D ; then, since each parallelogram 

 is double of the triangle B D C, * 

 Pr. 34. the parallelograms are 

 equal. 



Next, let the sides AD, EF 

 terminate in different points, D, E ; then A D = B C, 



t Pr. 34. and E F = B C,f . '. A D = E F, and D E is 

 A D t I A ED r 



common, .. the whole, or remainder, A E = the whole, o* 



Pr. 34. remainder, D F: also A B = D C,* . . A E, A B 

 t Pr. 29. = D F, D C, each to each ; also the angle A = 



F D C, t . . the triangle K A B = triangle F D C. Take 

 the triangle E A B from the trapezium A B C F, and 

 from the same trapezium take the equal triangle F D C : 

 the remainders must be equal ; that is, the parallelogram 

 E B C F= the parallelogram A B C D ; . ' . parallelograms 

 upon the same base, <fcc. Q. E. D. 



PROPOSITION XXXVI. THEOREM. 



Parallelograms (A B C D, E F G H) upon equal bnses 

 (B C, F G) and between the same parallels (A H, B G) 

 are equal. 



Draw B E, C H. Then because B C = F G, and F G 



= EH; .-. BC = 

 H EH; and these are 

 parallels, and joined 

 towards the same parts 

 by B E, C H, . . E B, 

 HC are equal and 

 Pr. 33. parallel,* 

 .-. EBCHisaparal- 

 t Pr. 4. lelogram, t 

 . Def. and it is= A B C D,* also E B C H= E K G 1 1 ,* 



Pr. ss. . . the parallelogram A B C D = K F G H, 

 . '. parallelograms upon equal bases, &o. Q. E. D. 



PROPOSITION XXXVII. THEOREM. 

 Triangles (A B C, D B C) on the same base. (B C) and 



between the same parallels (EF, B C) are equal. 

 This proposition U only a particular case of that which follows ; and 

 ft* the particular is not made use of in the more general demon- 

 stration, it may be omitted, as quite superfluous. Prop. XXXVI II. 



proves that triangles between the same parallels are equal, provided 

 only that their banes are equal, without any restriction as to 

 whether the bases coincide or not. whatever is proved as to equal 



things is, of course, proved when the thinfrs are not only equal, 

 but identical. 1'rop. XXXV. is. like the pri^ent, only a particular 

 case of that next in order, but the proof of the latter requires that 

 the particular case be previously established; so that, although. 

 Prop. XXXVI. really includes Prop. XXXV., yet Prop. XXXV. 

 mun not on that account be suppressed. 



The enunciation of the present useless proposition is retained here, 

 f-i'-lv in order that Euclid's subsequent propositions may not be 

 disturbed. 



PROPOSITION XXXVIII. THEOREM. 

 Triangles (A R C, D E F) upon equal bases (B C,E F) and 



between tlte same parallels (B F, G H) are equal. 

 G A a M Through B draw B G 



Pr. 31 . parallel to C A , * 

 and through F draw 

 KI parallel to Kl), 

 the lines thus drawn 

 terminating in G II. 

 ThunGBC.A, DKFIt 

 are parallelograms, f 

 t Pr. 34. and, being on equal bases, and between the 

 Def. same parallels, they are rqnal,' .'. their halves 

 Pr. 36. are equal ; that is, tlie triangle A B (2 = the tri- 

 angle D E F, .-. triangles upon equal banes, etc. Q. E. D. 



PROPOSITION XXXIX. THEOREM. 



Equal triangles (A B C, D B C) upon the same base (B C) 

 and on the same side of it, are between the same 

 parallels. 



This proposition, like Prop. XXXVII., is superfluous j it is included 

 in the next. 



PROPOSITION XL. THEOREM. 



Equal triangles (A B C, D E F) on the same side of the 

 same straight line, and having equal bases, are between 

 the same parallels. 

 Draw A D. Then if A D be not parallel to B F, let 



C 



