15. 1. BOOK I. but the angle AGH is equal to the angle EGB; therefore likewise EGB is equal to GHD; add to each of these the angle BGH; therefore the angles EGB, BGH are equal 13. 1. to the angles BGH, GHD; but EGB, BGH are equal to two right angles; therefore also BGH, GHD are equal to two right angles. Wherefore, if a straight, &c. Q.E. D. PROP. XXX. THEOR. STRAIGHT lines which are parallel to the same straight line are parallel to each other. 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 an29. 1. gle AGH is equal a to the anAgle GHF. Again, because the straight line GK cuts the parallel straight lines EF, CD, Ethe angle GHF is equal to the angle GKD; and it was shown that the angle AGK is equal to the angle GHF; C K -B H F -D therefore also AGK is equal to GKD; and they are alter-27.1. nate angles; therefore AB is parallel to CD. Wherefore straight lines, &c. Q. E. D. PROP. XXXI. PROB. To draw a straight line through a given point parallel to a given straight line. Let A be the given point, and BC the given straight line; it is required to draw a straight line through the point A, paral lel to the straight line BC. In BC take any point D, and join AD; and at the point A, in E B D C F * 23. 1. the straight line AD, make the angle DAE equal to the angle ADC; and produce the straight line EA to F. Because the straight line AD, which meets the two straight lines BC, EF, makes the alternate angles EAD, ADC equal to one another, EF is parallel to BC. There- Book I. fore the straight line EAF is drawn through the given point A parallel to the given straight line BC. Which was to be done. PROP. XXXII. THEOR. If a side of any 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 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. 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 B equal. Again, because 27. 1. a E 31. 1. D AB is parallel to CE, and BD falls upon them, the exterior angle ECD is equal to the interior and opposite angle ABC; 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 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 also the angles CBA, BAC, ACB, are equal to two right angles. Wherefore if a side of a triangle, &c. Q. E. D. COR. 1. All the interior angles of any rectilineal figure, together with four right angles, are equal to twice as many right angles as the figure has sides. For any rectilineal figure ABCDE, can be divided into as many triangles as the figure has sides, by drawing straight lines from a point F within the figure to A b 29. 1. c 13. 1. Book I. each of its angles. And, by the preceding proposition, all the angles of these triangles are equal to twice as many right angles as there are triangles, that is, as there are sides of the figure and the same angles are equal to the angles of the figure, together with the angles at the point F, which * 2 Cor. is the common vertex of the triangles: that is, together with four right angles. Therefore all the angles of the figure, together with four right angles, are equal to twice as many right angles as the figure has sides. 15. 1. Cor. 2. All the exterior angles of any rectilineal figure' are together equal to four right angles. Because every interior angle ABC, with its adjacent exteb 13. 1. rior ABD, is equal to two right angles; therefore all the interior together with all the exterior angles of the figure, are equal to twice as many right angles as there are sides D of the figure; that is, by the foregoing corollary, they are equal to all the interior angles of the figure, together with four right angles; therefore all the exterior angles are equal to four right angles. PROP. XXXIII. THEOR. THE straight lines which join the extremities of two equal and parallel straight lines towards the same parts, are also themselves equal and parallel. Let AB, CD be equal and parallel straight lines, and join-. ed towards the same parts by the straight lines AC, BD; AC, BD are also equal and parallel. Join BC; and because AB is parallel to CD, and BC meets them, the alternate an B 29. 1. gles ABC, BCD 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 4. 1. AC is equal to the base BD, and the triangle ABC to the triangle BCD, and the other angles to the other angles", each to each, to which the equal sides are opposite; there fore the angle ACB is equal to the angle CBD; and be- Book I. cause the straight line BC meets the two straight lines AC, BD, and makes the alternate angles ACB, CBD equal to one another, AC is parallel to BD; and it was shown to 27. 1. be equal to it. Therefore, straight lines, &c. Q. E. D. PROP. XXXIV. THEOR. THE opposite sides and angles of parallelograms are equal to one another, and the diameter bisects them, that is divides, them into two equal parts. N. B. A parallelogram is a four-sided figure, of which the opposite sides are parallel; and the diameter is the straight line joining two of its opposite angles. Let ACDB be a parallelogram, of which BC is a diameter; the opposite sides and angles of the figure are equal to one another; and the diameter BC bisects it. Because AB is parallel to A CD, and BC meets them, the alternate angles ABC, BCD are equal to one another; and because AC is parallel to BD, and BC meets them, the alternate angles ACB, CBD, are B 29. 1, equal a to one another; wherefore the two triangles ABC, CBD have two angles ABC, BCA in one, equal to two angles BCD, CBD in the other, each to each, and one side BC common to the two triangles, which is adjacent to their equal angles; therefore their other sides shall be equal, each to each, and the third angle of the one to the third angle of the other, viz. the side AB to the side CD, and 26. 1. AC to BD, and the angle BAC equal to the angle BDC: And because the angle ABC is equal to the angle BCD, and the angle CBD to the angle ACB, the whole angle ABD is equal to the whole angle ACD: And the angle BAC has been shown to be equal to the angle BDC; therefore the opposite sides and angles of parallelograms are equal to one another; also, their diameter bisects them; for AB being equal to CD, and BC common, the two AB, BC are equal to the two DC, CB, each to each; and the angle ABC is equal to the angle BCD; therefore the triangle Book I ABC is equal to the triangle BCD, and the diameter BC divides the parallelogram ACDB into two equal parts. Q. E.D. € 4.1. PROP. XXXV. THEOR. 1 See N. PARALLELOGRAMS upon the same base, and between the same parallels, are equal to one another. See the 2d Let the parallelograms ABCD, EBCF be upon the same and 3d fi- base BC, and between the same parallels AF, BC; the parallelogram ABCD shall be equal to the parallelogram EBCF. gures. 1. Ax. • 2 or 3 Ax. F But, if the sides AD, EF, opposite to the base BC of the parallelograms ABCD, EBCF, be not terminated in the same point; then, because ABCD is a parallelogram, AD is equal to BC; for the same reason EF is equal to BC; wherefore AD is equal to EF; and DE is common; therefore the whole, or the remainder, AE is equale to the whole, or the remainder DF; AB also is equal to DC; and the two EA, AB are therefore equal to DE AE DF B В e the two FD, DC, each to each; and the exterior angle 29.1. FDC is equald to the interior EAB, therefore the base EB 4. 1. is equal to the base FC, and the triangle EAB equal to the triangle FDC. Take the triangle FDC from the trapezium ABCF, and from the same trapezium take the tri13 Ax, angle EAB: the remainders therefore are equal, that is, the parallelogram ABCD is equal to the parallelogram EBCF. Therefore parallelograms upon the same base, &c. Q.E. D. |