terminated in 1. If the sides AD, DF, of the ms ABCD, = double the BDC, :: the 7m ABCD = 2. If the sides AD, EF, be not terminated in the same point (figs. 2 and 3), Prop. 33. Ax. 6. And ** Sthe whole or Take the remain. AE Prop. 33. } whole or remain. DF. Ax. 2, 3. EA, AB = FD, DC, ea. to ea. ext. FDC = int. EAB, base EB = base FC, Δ ΕΑΒ = Δ FDC. EAB from the trapezium ABCF; and from the same trapezium take the FDC; then the remainders are =; that is, ABCD: = EBCF. Wherefore parallelograms, &c. COR.-Triangles upon the same base, and between the same parallels, are equal to one another. For if the diameters AC, FB, be drawn (figs. 2 and 3), the As ABC, FBC, Prop. 33. Prop. 28. Prop. 4. Ax. 3. Ax. 7. are the halves of the equalms ABCD, EBCF, .. the As are equal to one another. PROP. XXXV. THEOR. 36. 1 Eu. Parallelograms upon equal bases, and between the same parallels, are equal to one another. Let ABCD, EFGH, be ms upon= bases BC, FG, and between the same ||s, AH, BG ; Prop. 34. Join BE, CH, BC= FG, EH = FG, BC= EH; str. lines EB, HC, join the extremities of = and || str. lines, they = and : that is, BE is = and | to CH, EBCH is a m EBCH = since they are on the same base BC and be tween the same ||s. For the same reason, EFGH = m EBCH, m ABCD = m EFGH. Ax. 1. Wherefore parallelograms, &c. COR. 1.-Triangles upon the equal bases and between the same parallels are equal to each other. For draw the diams. AC, EG; the As ABC, EFG, are the halves of the equal 'ABCD, EFGH, and are therefore equal Ax. 7. to each other. ms COR. 2.-If a parallelogram and a triangle be upon the same or equal bases, and between the same parallels, the parallelogram is double the triangle. For the 7m ABCD is double the ABC, or double the is its equal, by last Cor. EFG, which PROP. XXXVI. THEOR. 39. 1 Eu. Equal triangles upon the same base, and upon the same side of it, are between the same parallels. Let the = As ABC, DBC, be on the same base BC, and on the same side of it; then the As will be between the same ||s. ABC = DBC, :: Δ EBC =ΔDBC, i. e. less = greater, which is impossible; AEBC. (being on the same base BC, and be tween the same ||s BC, AE; In the same manner it may be proved that no other line but AD || BC, .. AD || BC. Wherefore equal triangles upon, &c. COR.--In the same way it may be shown, that equal triangles upon equal bases and towards the same parts, are between the same parallels PROP. XXXVII. THEOR. 43. 1 Eu. The complements of the parallelograms which are about the diameter of any parallelogram, are equal to each other. whole figure ABCD, are called complements. Then comp. BEKG = comp. KĤDF. which make up the |