PROPOSITION XIV.-THEOREM. If two triangles, ABC, DEF, have two sides, AB, BC, of the one respectively equal to, DE and EF, two sides of the other, but the angle ABC, included by the two sides of the one, greater than the angle DEF, included by the corresponding sides of the other; then the side AC is greater than the side DF. B E Let ABG be the part of the LABC, which is = DEF, and let BG be EF or BC. Then the As ABG, DEF, are equal in all respects, (Prop.5), and have the side AG= DF. And as BC and BG are, the LBGCis the LBCG (Prop.7); but the LBCG is the LACG, .. also the LBGC is the LACG, (Ax. 12); much more then is the LAGC the LACG, and hence (Prop. 12), the side AC is AG, and.. also its equal DF (Ax. 13). PROPOSITION XV.-THEOREM. Q. E. D. Д.Д. If two triangles, ABC, DEF, have the two sides AC, CB of the one respectively equal to two sides DF, FE of the other, but the remaining side AB of the one greater than the remaining side DE of the A other; the angle ACB will be greater than the angle DFE. BD For, if the LACB be not the ¿DFE, it must either be equal to it or less; the LACB is not = DFE, for then (Prop. 5), the base AB would be DE, which it is not; neither is the LACB the LDFE, for then (Prop. 14), the base AB would be the base DE, which it is not, .. the LACB is the LDFE. PROPOSITION XVI.-THEOREM. Q. E. D. straight lines AB and CD are parallel. and BGH be the two Ls BGH and GHD; but the two L8 EGB and BGH are together = two Ls (Prop. 1), .. the two Ls BGH and GHD are together equal to two right angles. Q. E. D. Cor. If two straight lines, LM and CD, being cut by a third, EF, make the two interior angles LGH and GHC, on the same side of the cutting line, together less than two right angles, these two lines will meet if produced far enough, on that side where the angles formed are less than two right angles. For if they do not meet on that side, they must either be, or they will meet being produced on the other side; they are not |, for then the two L/s LGH and GHC would be together two r's, which they are not: neither do they meet, being produced towards M and D, for then the two Ls MGH and GHD would be two Ls of a ▲, and .. less than two Ls (Prop. 10); but the four Ls LGH, GHC, MGH, and GHD, are together four Ls (Prop. 1), of which the two, LGH and GHC, are together less than wo Ls, the two, MGH and GHD, are two Ls, ence LM and CD do not meet towards M and D, and ince it has been shown that they are not, they must neet towards L and C. PROPOSITION XVIII.—THEorem. Through a point A to draw straight line parallel to a given traight line BC. In BC take any point D, join D, and at the point A make Cor. 2, Prop. 9), the [DAE = Q. E. D. A E F B D le LADC, and produce EA to F, the LEAD = the ADC, and they are alternate angles, . (Prop. 16), EF to BC, hence through the point A, a straight line, CF, has been drawn || to BC. PROPOSITION XIX.-THEOREM. If a side BC of a riangle ABC be proluced to D, the extefor angle ACD will be equal to the two interior angles CBA and BAC; and the three interior angles of B Q. E. D. every triangle are together equal to two right angles. E ACE are Through C draw (Prop. 18), CE || to AB. Then AB is | to CE, and AC meets them, the alternate s BAC and (Prop. 17); and AB is || to CE, and DB falls upon them, the exterior [ECD is the interior ABC, but the ACE is the LBAC, .. the whole ACD is the two Ls CAB and ABC. To each of these equals add the LACB,.. the two s ACD, ACB are the three Дs CAB, ABC, and BCA, but the two Ls ACD, ACB are together two Ls, .. the three angles CAB, ABC, and BCA are together equal to two right angles. Cor. 1. If two angles in one triangle be equal to two angles in another triangle, the remaining angles of those triangles are equal. Cor. 2. If one angle in a triangle be equal to an angle in another triangle, the sum of the remaining angles in each triangle are equal. Cor. 3. If one angle in a triangle be a right angle, the other two angles are together equal to a right angle; and hence each of them is an acute angle. Cor. 4. Every triangle has at least two acute angles. Cor. 5. Hence from this proposition, and (Prop. 6), if two triangles have two angles in the one equal to two angles in the other, and a corresponding side equal in each, they are equal in all respects. PROPOSITION XX.-THEOREM. If two lines AB and BC meeting in a point B, be respectively parallel to DE and EF meeting in a A point E, the included B E angles ABC and DEF are equal. For join B, E, and produce BE to G. Since AB is || to DE, and GB falls on them, the GBA is the [DEG, (Prop. 17); for a like reason the CBG is .. taking equals from equals, there remains the the /DEF. the FEG, ABC = Q. E. D. PROPOSITION XXI.-THEOREM. If all the sides of a rectilineal figure be produced, the sum of all the exterior angles so formed will be together equal to four right angles. Le'; .. the sum of all the Ls a, b, c, d, e, are equal to the e E sum of all the Ls a', b', c', d', and e; but the Ls a', b', c', d', and e' are together equal to four Ls (Prop. 1, Cor. 3), .. all the exterior Ls a, b, c, d, and e, are together equal to four right angles. Q. E. D. PROPOSITION XXII.-THEorem. All the interior angles of any rectilineal figure are together equal to twice as many right angles as the figure has sides, wanting four right angles. For (Figure to Prop. 21) every interior LEAB, together with its adjacent exterior LBAK, are together equal to two Ls; all the interior, together with all the exterior, are equal to twice as many rLs as the figure has sides; but all the exterior Ls are four Ls, (Prop. 21); .. all the interior are twice as many right angles as the figure has sides, wanting four right angles. Q. E. D. Cor. 1. All the interior angles of any quadrilateral figure are together equal to four right angles. Cor 2. If the sum of two angles of a quadrilateral figure be equal to two right angles, the sum of the remaining angles is also equal to two right angles. PROPOSITION XXIII.-THEOREM. Of all straight lines, drawn from the point A to the straight line BC, the perpendicular AD is the least; AE, which is nearer to the perpendicular, is less than AF, which is more remote, and there can only be drawn two |