a equal to DE, and BC to EF, the two fides GB, BC, are equal Book 1. to the two DE, EF, each to each; and the angle GBC is equal to the angle DEF; therefore the bafe GC is equal to the bafe a 4. 1. DF, and the triangle GBC to the triangle 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 to the bafe DF, and the third angle BAC to the third angle EDF. Next, let the fides A which are oppofite to B a 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 bafe AH is equal to the bafe DF, and the triangle ABH to the triangle DEF, and the other angles fhall 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 alfo 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 opposite angle BCA; which is impoffible; wherefore BC is not unequal to EF, that is, b 16.1. 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 base AC is equal to the bafe DF, and the third angle BAC to the third angle EDF. Therefore, if two triangles, &c. Q. E. D. 34 Book 1. 416. 1. IF PROP. XXVII. THEOR. a ftraight line falling upon two other straight lines makes the alternate angles equal to one another, thefe two straight 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. A E For, if it be not parallel, AB and CD being produced fhall 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 than the interior and oppofite angle EFG; but it is alfo 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 they do not meet towards A, C; but thofe ftraight lines which meet neither way, though C B G F D b35. Def. produced ever fo far, are parallel b to one another. AB therefore is parallel to CD. Wherefore, if a ftraight line, &c. Q. E D. PROP. XXVIII. THEOR. Fa 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 inakes the interior angles upon the fame fide together equal to two right angles; the two ftraight lines fhail be parallel to one angle EGB equal to the angle AGH, the angle AGH is equal Book 1. to the angle GHD; and they are the alternate angles; therefore AB is parallel to CD. Again, because the angles BGH, GHD a 15. 1. are equal to two right angles, and that AGH, BGH are alfo b 27. 1. c By Hyp, equal 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. PROP. XXIX. THEOR. notes on fition. IF a straight fre falls upon two parallel ftraight lines, it See the makes the alternate angles equal to one another; and this propothe exterior angle equal to the interior and oppofite upon the fame fide; and likwife 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, GHD; and the two interior angles BGH, GHD upon the fame fide are together equal to two right angles. E A For, if AGH be not equal to GHD, one of them muft be greater than the other; let AGH be the greater; and C because the angle AGH is greater than the angle GHD, add to each of a G B them the angle BGH; therefore the angles AGH, BGH are greater than the angles BGH, GHD; but the angles AGH, BGH are equal to two right angles; therefore the angles a 13. I. BGH, GHD are lefs than two right angles; but thofe ftraight lines which, with another straight line falling upon them, make the interior angles on the fame fide lefs than two right angles, do meet together if continually produced; therefore the 12. ax. ftraight lines AB, CD, if produced far enough, thall meet; but See the they never meet, fince they are parallel by the hypothefis; therefore the angle AGH is not unequal to the angle GHD, that is, it is equal to it; but the angle AGH is equal to the bis, & angle EGB; therefore likewife EGB is equal to GHD; add to C 2 b each notes on this prope P c Book I. each of these the angle BGH; therefore the angles EGB, BGH are equal to the angles BGH, GHD; but EGB, BGH are equal to two right angles; therefore alfo BGH, GHD are equal to two right angles. Wherefore, if a straight line, &c. Q. E. D. 13. I. a 29. I. b 27.1. I. PROP. XXX. THEOR.. TRAIGHT lines which are parallel to the fame ftraight line are parallel to one another. ST Let AB, CD be each of them parallel to EF; AB is also parallel to CD. Let the ftraight line GHK cut AB, EF, CD; and because GHK cuts the parallel ftraight a E lines AB, EF, the angle AGH 2 C therefore AB is parallel to CD. Wherefore ftraight lines, &c. Q.E. D. PROP. XXXI. PRO B. O draw a ftraight line through a given point parallel to a given straight line. E A F Let A be the given point, and BC the given straight line; it In BC take any point D, and join a a 23. 1. ftraight line AD make the angle B DAE equal to the angle ADC; and produce the ftraight line EA to F. b 27. I. D Because the straight line AD, which meets the two ftraight lines BC, EF, makes the alternate angles EAD, ADC equal to one another, EF is parallel to BC. Therefore the ftraight line EAF EAF is drawn through the given point A parallel to the given Book I. ftraight line BC. Which was to be done. IF a fide of any triangle be produced, the exterior angle is equal to the two interior and oppofite angles; and the three interior angles of every triangle are equal to two right angles. Let ABC be a triangle, and let one of its fides BC be produced to D; the exterior angle ACD is equal to the two interior and oppofite angles CAB, ABC; and the three interior angles of the triangle, viz. ABC, BCA, CAB are together equal to two right angles. Through the point C draw CE parallel to the straight line AB; and because AB is parallel to CE and AC meets them, the alternate angles BAC, ACE are equal b. Again, because AB is parallel to CE, and BD falls upon them, the exterior angle ECD B is equal to the interior and A a 31. I. b 29. I. D oppofite angle ABC; but the angle ACE was fhown to be equal to the angle BAC; therefore the whole exterior angle ACD is equal to the two interior and oppofite angles CAB, ABC; to these equals add the angle ACB, and the angles ACD, ACB are equal to the three angles CBA, BAC, ACB; but the angles ACD, ACB are equal to two right angles; therefore alfo the c 13. I. angles CBA, BAC, ACB are equal to two right angles. Wherefore, if a fide of a triangle, &c. Q. E. D. c |