Let ABC, DEF be two triangles which have the two fides Book I, AB, AC equal to the two DE, DF, each to each, viz. AB equal to DE, and AC to DF; but the angle BAC greater than the angle EDF; the bafe BC is alfo greater than the base EF. Of the two fides DE, DF, let DE be the fide which is not greater than the other, and at the point D, in the straight line DE, make a the angle EDG equal to the angle BAC,; and 1 23. I. make DG equal b to AC or DF, and join EG, GF. Because AB is equal to DE, and AC to DG, the two fides BA, AC are equal to the two ED, DG, each to each, and the a b 3. I. e C 4. I. ds. i. angle DFG is greater than EGF; and much more is the angle EFG greater than the angle EGF; and because the angle EFG of the triangle EFG is greater than its angle EGF, and that the greater e fide is oppofite to the greater angle; the fide EG € 19. I. is therefore greater than the fide EF; but EG is equal to BC; and therefore alfo BC is greater than EF. Therefore, if two triangles, &c. Q. E. D. PROP. XXV. THEOR. two triangles have two fides of the one equal to two fides of the other, each to each, but the base of the one greater than the bafe of the other; the angle alfo contained by the fides of that which has the greater base, shall be greater than the angle contained by the fides equal to them, of the other. Let ABC, DEF be two triangles which have the two fides AB, AC equal to the two fides DE, DF, each to each, viz. AB equal to DE, and AC to DF; but the base CB is greater than the base EF; the angle BAC is likewise greater than the angle EDF. For, Book 1. For, if it be not greater, it must either be equal to it, or lefs; but the angle BAC is not equal to the angle EDF, because then the base BC would is not less than the angle EDF; and it was fhewn that it is not equal to it; therefore the angle BAC is greater than the angle EDF. Wherefore, if two triangles, &c. Q. E. D. PROP. XXVI. THEOR. F two triangles have two angles of one equal to two angles of the other, each to each; and one fide equal to one fide, viz. either the fides adjacent to the equal angles, or the fides oppofite to equal angles in each; then fhall the other fides be equal, each to each; and alfo the third angle of the one to the third angle of the other. D Let ABC,DEF be two triangles which have the angles ABC, BCA equal to the angles DEF, EFD, viz. ABC to DEF, and BCA to EFD; alfo one fide equal to one fide; and first let thofe fides be equal which are adjacent to the angles that are equal in the two triA angles; viz. BC to EF; the other fides fhall be equal, each to each, viz. AB to DE, and AC to DF; and the third angle BAC to the third greater of the two, and make BG equal to DE, and join GC; therefore, because BG is equal to DE, and BC to EF, the two fides fides GB, BC are equal to the two DE, EF, each to each; and Book I. the angle GBC is equal to the angle DEF; therefore the bafe GC is equal a to the bafe DF, and the triangle GBC to the tri- a 4. I. angle DEF, and the other angles to the other angles, each to each, to which the equal fides are oppofite; therefore the angle GCB is equal to the angle DFE; but DFE is, by the hypothefis, equal to the angle BCA; wherefore alfo the angle BCG is equal to the angle BCA, the lefs to the greater, which is impoffible; therefore AB is not unequal to DE, that is, it is equal to it; and BC is equal to EF; therefore the two AB, BC are equal to the two DE, EF, each to each; and the angle ABC is equal to the angle DEF; the base therefore AC is equal a to the base DF, and the third angle BAC to the third angle EDF. Next, let the fides which are oppofite to A equal angles in each triangle be equal to one another, viz. AB to DE; likewise in this cafe, the other fides fhall be equal, AC to DF, and BC to EF; and also the D AA B third angle BAC to the third EDF. HC E For, if BC be not equal to EF, let BC be the greater of them, and make BH equal to EF, and join AH; and because BH is equal to EF, and AB to DE; the two AB, BH are equal to the two DE, EF, each to each; and they contain equal angles; therefore the base AH is equal to the bafe DF, and the triangle ABH to the triangle DEF, and the other angles shall be equal, each to each, to which the equal fides are oppofite; therefore the angle BHA is equal to the angle EFD ; but EFD is equal to the angle BCA; therefore also the angle BHA is equal to the angle BCA, that is, the exterior angle BHA of the triangle AHC is equal to its interior and oppofite angle BCA ; which is impoffible b; wherefore BC is not unequal to EF, b 16. 1. that is, it is equal to it; and AB is equal to DE; therefore the two AB, BC are equal to the two DE, EF, each to each; and they contain equal angles; wherefore the bafe AC is equal to the base DF, and the third angle BAC to the third angle EDF. Therefore, if two triangles, &c. Q. E. D. Book I. a 16. 1. PROP. XXVII. THEOR. F a fraight line falling upon two other ftraight lines makes the alternate angles equal to one another, thefe two ftraight lines fhall be parallel." Let the ftraight line EF, which falls upon the two straight lines AB, CD make the alternate angles AEF, EFD equal to one another; AB is parallel to CD. For, if it be not parallel, AB and CD being produced shall meet either towards B, D, or towards A, C; let them be produced and meet towards B, D in the point G; therefore GEF is a triangle, and its exterior angle AEF is greater a than the interior and oppofite angle EFG; but it is also equal to it, which is impoffible; therefore AB and CD being produced do not meet towards B, D. In like manner it may be demonftrated that C they do not meet towards A, C; but those straight lines A E B G F D which meet neither way, though produced ever so far, are pa b 35. Def. rallel b to one another. AB therefore is parallel to CD. Wherefore, if a straight line, &c. Q. E. D. PROP. XXVIII. THEOR. F a ftraight line falling upon two other ftraight lines makes the exterior angle equal to the interior and oppofite upon the fame fide of the line; or makes the interior angles upon the fame fide together equal to two right angles; the two ftraight lines fhall be parallel to one another. Let the ftraight line EF, which falls upon the two ftraight lines AB, CD, make the exterior angle EGB equal to the interior and A oppofite angle GHD upon the fame fide; or make the interior angles on the fame fide FGH, C GHD together equal to two right angles; AB is parallel to CD. Because the angle EGB is equal to the angle GHD, and the E G -B -D H F angle C a Book I. n b 27. I. angle EGB equal a to the angle AGH, the angle AGH is equal to the angle GHD; and they are the alternate angles; therefore AB is parallel b to CD. Again, because the angles BGH, GHD 15. I. are equal to two right angles; and that AGH, BGH, are also c By Hyp. equal d to two right angles; the angles AGH, BGH are equal d 13. 1. to the angles BGH, GHD: Take away the common angle BGH; therefore the remaining angle AGH is equal to the remaining angle GHD; and they are alternate angles; therefore AB is parallel to CD. Wherefore, if a straight line, &c. Q E. D. I PROP. XXIX. THEOR. this Fakeage alternate and notes onpoa straight line fall upon two parallel ftraight lines, it See the makes the alternate angles equal to one another; the exterior angle equal to the interior and oppofite upon fition. the fame fide; and likewife the two interior angles upon the fame fide together equal to two right angles. Let the ftraight line EF fall upon the parallel ftraight lines AB, CD; the alternate angles AGH, GHD are equal to one another; and the exterior angle EGB is equal to the interior and oppofite, upon the fame fide, E GHD; and the two interior angles BGH, GHD upon the fame fide. are together equal to two right angles. For, if AGH be not equal to GHD, one of them must be greater C than the other; let AGH be the greater; and because the angle AGH is greater than the angle GHD, add G H B D F to each of them the angle BGH; therefore the angles AGH, BGH are greater than the angles BGH, GHD; but the angles * a See the AGH, BGH are equal to two right angles; therefore the a 13. I. angles BGH, GHD are lefs than two right angles; but those ftraight lines which, with another ftraight line falling upon them, make the interior angles on the fame fide less than two right angles, do meet together if continually produced; therefore 12. ax, the ftraight lines AB, CD, if produced far enough, fhall meet ; notes on but they never meet, fince they are parallel by the hypothefis; this propotherefore the angle AGH is not unequal to the angle GHD, that fition. is, it is equal to it; but the angle AGH is equal b to the angle b 15. 1. EGB; therefore likewife EGB is equal to GHD; add to each C 2 of |