a. 16. 1. Book 1. PROP. XXVII. THEOR. W IF a straight line falling upon two other straight lines makes the alternate angles equal to one another, these two straight lines shall be parallel. Let the straight line EF which falls upon the two straight lines AB, CD make the alternate angles AEF, EFD equal to one another ; AB is parallel tɔ CD. For if it be not parallel, AB and CD being produced shall meet either towards BD or towards AC. let them be produced and meet towards BD in the point G; therefore GEF is a triangle, and its exterior angle AEF is greater than the interior and opposite angle EFG; but it is also equal to it, which is impossible. there fore AB and CD being pro- A E. B duced do not meet towards BD. in like manner it may be demonstrated that they do not CF D meet towards AC. but those straight lines which meet neib. 35. Der ther way tho' produced ever so far are parallel b to one another. AB therefore is parallel to CD. wherefore if a straight line, &c. Q.E.D. PROP. XXVIII. THEOR. makes the exterior angle equal to the interior and opposite upon the same side of the line; or makes the interior angles upon the same side together equal to two right angles; the two straight lines shall be parallel to one another. Let the straight line EF which falls upon the two straight lines AB, CD make the exterior angle E EGB equal to the interior and opposite angle GHD upon the fame fide; or make the interior angles A -B H F EGB equal to the angle AGH, the angle AGH is equal to the Book I. angle GHD; and they are the alternate angles; therefore AB is parallel 6 to CD. again, because the angles BGH, GHD are equal “a. 15. 1. to two right angles, and that AGH, BGH are also equal to twob.27. 1. right angies ; the angles AGH, BGH are equal to the angles BGH, .. 13.. GHD. take away the common angle BGH, therefore the remaining angle ACH 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. IF. on PRO P. XXIX. THEO R. a straight line falls upon two parallel straight lines, See the it makes the alternate angles equal to one another ; this Propoand the exterior angle equal to the interior and opposite lition. up in the same lide; and likewise the two interior angles upon the same lide together equal to two right angles. Let the straight line EF fall upon the parallel straight lines AB, CD. the alrernate angles AGH, GHD are equal to one another ; and the exterior angle EGB is equal to the interior and opposite, upon the same side, GHD; and the two interior angles BGH, GHD upon E the same fide are together equal to two right angles. A G B For if AGH be not equal to GHD, one of them must be greater than theother; let AGH be the greater and be-C H D cause the angle AGH is greater than F the angle GHD, add to each of them the angle BGH; therefore the angles AGH, BGH are greater than the angles BGH, GHD. but the angles AGH, BGH are equal to a. 13. 1. two right angles; therefore the angles BGH, GHD are less than two right angles. but those straight lines which with another straight line falling upon them make the interior angles on the same side less than two right angles, do meet * together if continually pro- . 12. Ax. duced ; therefore the straight lines AB, CD if produced far enough fall meet. but they never meet, since they are parallel by the Hy- this Propopothesis. therefore the angle AGHis not unequal to the angle GHD, Gtion. that is, it is equal to it. but the angle AGH is equal b to the angle b. 15. $. EGB; therefore likewise EGB is equal to GHD. add to each of See the notes on Book 1 these the angle BGH, therefore the angles' EGB, BGH are equal to the angles BGH, GHD; but EGB, BGH are equal' to two $13. s. right angles ; therefore also BGH, GHD are equal to two right angles. wherefore if a straight line, &c. Q. E. D. PRO P. XXX. THEOR. STRAIGHT lines which are parallel to the fame 2. 29. I. straight line, are parallel to one another. Let AB, CD be each of them parallel to EF ; AB is also parallel to CD. Let the straight line GHK cut AB, EF, CD; and because GHK cuts the parallel straight lines AB, EF, the angie AGH is equal é to the angle GHF. again, because the G straight line GK cuts the parallel A B straight lines EF, CD, the angle GHF is equal to the angle E HI F GKD. and it was shewn that the c. K D angle AGK is equal to the angle GHF; therefore alfo AGK is equal to GKD, and they are alternate angles; therefore AB is parallel b.to CD. wherefore straight lines, &c. Q. E. D. To PRO P. XXXI. PRO B. to a given straight line. E A F In BC take any point D, and join o qual to the angle ADC; and produce the straight line EA to F. Because the straight line AD which meets the two straight lines BC, ET, makes the alternate angles EAD, ADC equal to one B. 27. 1. another, EF is parallel b to BC. therefore the straight line EAP is drawn thro' the given point A parallel to the given straight line Book I. BC. Which was to be done. 29. 1. PROP. XXXII. THEO R. angle is equal to the two interior and opposite 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 opposite angles CAB, ABC; and the three interior angles of the triangle, viz. ABC, BCA, CAB are together equal to two right angles. Thro' the point C draw CE parallel • to the straight line AB, a. 31. 1. and because AB is parallel to CE, and AC meets them, the alternate angles BAC, ACE are E b. 29. equal 6. again because AB is parallel to CE, and BD falls upon them, the exterior angle ECD is equal to the interior and opposite angle ABC. butB C D Cor. 1. All the interior angles E F А B 151. Book I. angles. And, by the preceding Proposition, all the angles of these triangles are cqual to twice as many right angles as there are triangles, that is, as there are sides of the figure. and the fame angles are equal to the angles of the figure, together with the angles at a. 2. Cor. the point F which is the common Vertex of the triangles ; that isa, together with four right angles. Therefore all the angles of the figurs, together with four right angles, are equal to twice as many right angles as the figure has sides. Cor. 2. All the exterior angles of any rectilineal figure are tow gether equal to four right angles. Because every interior angle ABC with its adjacent exterior b. 13. 1. ABD is equal b to two right angles; therefore all the interior to AY C foregoing Corollary, they are equal DB to all the interior angles of the figure, together with four right angles. therefore all the exterior angles are equal to four right angles. PRO P. XXXIII. THEOR. equal and parallel straight lines, towards the fame parts, are also themselves equal and parallel. Let AB, CD be equal and parallel straight lines, and joined to wards the fame parts by the straight A B Join BC, and because AB is pa- С are equal, and because AB is equal to CD, and BC common to the two triangles ABC, DCB, the two sides AB, BC are equal to the two DC, CB; and the angle ABC is equal to the angle BCD; therefore the base AC is equal o to the bafe BD, and the triangle ABC to the triangle BCD, and the other angles to the other an |