33 Wherefore AE is equal to the corresponding side CE, and BE to DE. Cor. Hence the diagonals of a rectangle are equal to each other; for if the angles at A and B were right angles, the triangles DAB and CBA would be equal (I. 3.) and consequently the base DB equal to AC. PROP. XXXII. THEOR. Lines parallel to the same straight line, are parallel to each other. If the straight line AB be parallel to CD, and CD pa rallel to EF; then is AB parallel to EF. For let the straight line GH cut these lines. G A C E I B K D LF Because AB is parallel to CD, the exterior angle GIA is equal (I. 25.) to the interior GKC; and since CD is parallel to EF, this angle GKC is, for the same reason, equal to GLE. Therefore GIA is equal to GLE, and consequently AB is parallel to EF (I. 25. Cor.) PROP XXXIII. THEOR. H Straight lines drawn parallel to the sides of an angle, contain an equal angle. If the straight lines AB, AC be parallel to DE, DF; the angle BAC is equal G B to EDF. A E For draw the straight line GAD through the vertices. And because AC is paral- D C 1 lel to DF, the exterior angle GAC is F (I. 25.) equal to GDF; and for the same reason, GAB is equal to GDE; there consequently re mains the angle BAC equal to EDF. C PROP. XXXIV. THEOR. An exterior angle of a triangle is equal to both its opposite interior angles, and all the interior angles of a triangle are together equal to two right angles. The exterior angle BCD, formed by the production of the side AC of the triangle ABC, is equal to the two opposite interior angles CAB and CBA, and all the interior angles CAB, CBA, and BCA of the triangle, are together equal to two right angles. B For through the point C draw (I. 26.) the straight line CE parallel to AB. And because AB is parallel to CE, the interior angle BAC is (I. 25.) equal to the exterior one ECD; and for the same reason the alternate angle ABC is equal to BCE. Wherefore the two angles CAB and ABC are equal to DCE and ECB, or to the whole exterior angle BCD. Add the adjacent angle BCA to each; and all the interior angles of the triangle ABC are together equal to the angles BCD and A BCA on the same side of the straight line AD, that is, to two right angles. N C ע Cor. 1. Hence the two acute angles of a right angled triangle are together equal to one right angle; and hence each angle of an equilateral triangle is two third parts of a right angle. Cor. 2. Hence if a triangle have its exterior angle, and one of its opposite interior angles, double of those of another triangle; its remaining opposite interior angle will also be double of the corresponding angle of the other. PROP. XXXV. THEOR. The angles round any rectilineal figure are together equal to twice as many right angles, abating four from the result, as the figure has sides. C For assume any point O within the figure, and draw straight lines OA, OB, OC, OD, and OE, to the several angular points. It is obvious, that the figure is thus resolved into as many triangles as it has B sides, and whose collected angles must be, therefore, equal to twice as many right angles. But the angles at the bases of these triangles constitute the internal angles of the figure. Consequently, from the whole amount there is to be deducted the vertical angles about the point O, and which are (Def. 4.) equal to four right angles. E A D Cor. Hence all the angles of a quadrilateral figure are equal to four right angles, those of a pentelateral figure equal to six right angles, and so forth; increasing the amount by two right angles for each additional side. PROP. XXXVI. THEOR. The exterior angles of a rectilineal figure are together equal to four right angles. The exterior angles DEF, CDG, BCH, ABI, and EAK of the rectilineal figure ABCDE are taken together equal to four right angles. : For each exterior angle DEF, with its adjacent inte rior one AED, is equal to two Cor. If the figure has a re-entrant angle BCD, the angle BCK which occurs in place of PROP. XXXVII. THEOR. If the opposite angles of a quadrilateral figure be equal, its opposite sides will be likewise equal and parallel. In the quadrilateral figure ABCD, let the angle at B be equal to the opposite one at D, and the angle at A equal to that at C; the sides AB, BC are equal and parallel to DC and DA. 1 For all the angles of the figure being equal to four right equal to two right angles, and the lines AB and CD (Cor. I. 25.) parallel; for the same reason, ABC and BAD being together equal to two right angles, the sides BC and AD, which limit them, are parallel. Cor. Hence a rectangle, or right-angled quadrilateral figure, has its opposite sides equal and parallel. PROP. XXXVIII. PROB. To draw a perpendicular from the extremity of a given straight line. From the point B, to draw a perpendicular to AB, with Because the triangles CDB and DBE are equilateral, the angles CBD and DBE are each of them equal to two third parts of a right angle (I. 34. cor.); and the triangles BDF, BEF, having the sides BD, DF equal to BE, EF, and the side BF common, are (I. 2.) equal, and con |