fore the base AH is equal to the base DF, and the triangle ABH to the triangle DEF, and the other angles to the other angles, each to each, to which the equal sides are opposite; therefore the angle AHB is equal to the angle DFE: But the angle DFE is equal to the angle ACB (Hyp.); therefore also the angle AHB is equal to the angle ACB, that is, the exterior angle AHB of the triangle AHC is equal to its interior and opposite angle ACB-which is impossible (1. 16): Therefore BC is not unequal to EF, that is, it is equal to it; and AB is equal to DE (Hyp.); therefore the two, AB, BC, are equal to the two, DE, EF, each to each, and they contain equal angles-therefore the base AC is equal to the base DF, and the third angle BAC to the third angle EDF. Wherefore, If two triangles &c. Q. E. D. PROP. XXVII. THEOR. 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. A E B G 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 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: Then GEF is a triangle, and its exterior angle AEF is greater than the interior and opposite angle EFG (1. 16): (Hyp.)—which is impossible; being produced do not meet C F D But it is also equal to it therefore AB and CD towards B, D: And, in like manner it may be demonstrated that they do not meet towards A, C: But (Def. 35) those straight lines which, being produced ever so far both ways, do not meet, are parallel to one another: Therefore AB is parallel to CD. Wherefore, If a straight line &c. Q. E. D. PROP. XXVIII. THEOR. If a straight line, falling upon two other straight lines, 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. E G B Let the straight line EF, which falls upon the two straight lines AB, CD, make the exterior angle EGB equal to the interior and opposite angle GHD upon the same side: or make the interior angles on the same side BGH, GHD together equal to two right angles: AB shall be parallel to CD. Because the angle EGB is equal to the angle GHD, and is also equal to the angle AGH (1. 15), the angle AGH is equal to the angle GHD: And they are alternate angles; therefore AB is parallel to CD (1. 27). Again, because the angles BGH, GHD are equal to two right angles, and that AGH, BGH are also equal to two right angles (1. 13), the angles AGH, BGH are equal 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. If a straight line fall upon two parallel straight lines, it makes the alternate angles equal to one another, and the exterior angle equal to the interior and opposite upon the same side, and also the two interior angles upon the same side together equal to two right angles. E Let the straight line EF fall upon the parallel straight lines AB, CD: it makes the alternate angles AGH, GHD, equal to one another, AG and the exterior angle EGB equal to the interior and opposite upon the same side, B D F GHD, and the two interior angles upon the same side, BGH, GHD, together equal to two right angles. For, if AGH be not equal to GHD, one of them must be greater than the other; let AGH be the greater: Then, because the angle AGH is greater than the angle GHD, add to each 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 (1. 13); therefore the angles BGH, GHD, are less than two right angles: But if a straight line falls upon two straight lines, so as to make the two interior angles on the same side of it together less than two right angles, these straight lines, being produced, shall meet (Ax. 12); therefore the straight lines AB, CD, if produced far enough, shall meet-but they never meet, since they are parallel (Hyp.): 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 angle EGB (1. 15); therefore also the angle EGB is equal to the angle GHD: add to each the angle BGH; therefore the angles EGB, BGH, are equal to the angles BGH, GHD: But the angles EGB, BGH, are equal to two right angles (1. 13); therefore also the angles BGH, GHD, are equal to two right angles. Wherefore, If a straight line &c. Q. E.D. PROP. XXX. THEOR. Straight lines, which are parallel to the same straight line, are parallel to one another. Let AB, CD be each of them parallel to EF: AB is parallel to CD. A C K B H F -D Let the straight line GHK cut AB, EF, CD: Then because GHK cuts the parallel straight lines AB, EF, the angle AGH is equal to the angle GHF (1.29): Again, be- Ecause the straight line GHK cuts the parallel straight lines EF, CD, the angle GHF is equal to the angle GKD (1.29): And it was shewn that the angle AGH is equal to the angle GHF; therefore also the angle AGK is equal to the angle GKD: And they are alternate angles; therefore (1. 27) AB is parallel to CD. Wherefore, Straight lines &c. Q. E.D. PROP. XXXI. PROB. Through a given point to draw a straight line parallel to a given straight line. Let A be the given point, and BC the given straight line: it is required to draw through the point A a straight line parallel to BC. E A F In BC take any point D, and join AD; and at the point A, in the straight line AD, make the angle DAE equal to the angle ADC; and produce the straight line EA to F. B D Because the straight line AD, falling upon the two straight lines EF, BC, makes the alternate angles EAD, ADC equal to one another, therefore EF is parallel to BC (1. 27): Wherefore the straight line EAF is drawn through the given point A parallel to the given straight line BC. Q. E.F. If any side of a triangle be produced, the exterior angle is equal to the two interior and opposite 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 sides BC be produced to D: the exterior angle ACD shall be equal to the two interior and opposite angles CAB, ABC; and the three interior angles of the triangle, BAC, ACB, CBA, are together equal to two right angles. B с D Through the point C draw CE parallel to the straight line AB (1. 31): Then because AB is parallel to CE, and AC meets them, the alternate angles BAC, ACE are equal (1. 29): Again, because AB is parallel to CE, and BD falls upon them, the exterior angle DCE is equal to the interior and opposite angle CBA (1.29): But the angle ACE was shown to be equal to the angle BAC; therefore the whole exterior angle ACD is equal to the two interior and opposite angles BAC, CBA: To each of these equals add the angle ACB; therefore the angles ACD, ACB, are equal to the three angles BAC, ACB, CBA: But the angles ACD, ACB are equal to two right angles (1.13); |