each to each, to which the equal fides are opposite. therefore the Book I. angle ACB is equal to the angle CBD, and becaufe the straight linem BC meets the two straight lines AC, BD and makes the alternate angles ACB, CBD equal to one another, AC is parallel to BD, c. 27. 1. and it was shewn to be equal to it. therefore straight lines &c. Q. E. D. 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 ibe opposite fides are parallel. and the diameter is the straight line joining two of its opposite angles. Let ABCD 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 CD, and BC meets them, the alternate angles ABC, BCD are equal to B one another, and because AC is parallel to BD, and BC meets them, the alternate angles ACB, CBD are equal o to one another. wherefore the two triangles ABC, C с D 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 5, viz. the side AB to the side CD, and AC to BD, and the b.26. 2. 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 shewn 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; there C. 4. 1. Book 1. therefore the triangle ABC is equal to the triangle BCD, and the diameter BC divides the parallelogram ACDB into two equal parts. PROP. XXXV. THEOR. the same parallels, are equal to one another, See the ad Let the parallelograms ABCD, EBCF be upon the same base BC and 3d Fi-and between the same parallels AF, BC. the pa. allelogram ABCD gures. shall be equal to the parallelogram EBCF. If the sides AD, DF of the parallelograms ABCD, DBCF oppo- of the parallelograms is double of the 2. 34. I. triangle BDC; and they are therefore But if the sides AD, EF opposite to BC; for the same reason, EF is equal to BC; wherefore AD is 6. 1. Ax. equal 6 to EF; and DE is common; therefore the whole, or the c. 2. or 3. remainder, AE is equal to the whole, or the remainder DF; AB also is equal to DC; and the two EA, AB are therefore equal to A DE F A E DF F 2 Ax, B B С base FC, and the triangle EAB equal to the triangle FDC. take trapezium take the triangle EAB; the remainders therefore are f. 3. Ax. equalf, that is, the parallelogram ABCD is equal to the paralle logram EBCF. therefore parallelograms upon the same base &c. PROP 6. 4. 1. THEOR. PROP. XXXVI. Book I. PARALLELOGRAMS upon equal bases and between the same parallels, are equal to one another. Let ABCD, EFGH be parallelograms upon equal bases BC, FG, and between the same A D E H parallels AH, BG; the parallelogram ABCD is equal to EFGH. Join BE, CH; and becáuse BC is equal to FG, and FG to EH, BC is B a. 34. # equal to EH; and they are parallels, and joined towards the same parts by the straight lines BE, CH. but straight lines which join equal and parallel straight lines towards the same parts, are themselves equal and parallel b; there. 6. 33. to fore EB, CH are both equal and parallel, and EBCH is a parallelogram; and it is equal to ABCD, because it is upon the same base c. 15. to BC, and between the fame parallels BC, AD. for the like reason the parallelogram EFGH is equal to the same EBCH. therefore also the parallelogram ABCD is equal to EFGH. Wherefore parallelograms, &c. Q. E. D. PROP. XXXVII. THEOR. same parallels, are equal to one another. E A D F BC. the triangle ABC is equal to the triangle DBC. Produce AD both ways to the points E, F, and thro’B a. 31. draw * BE parallel to CA; and thro' C draw CF parallel B C to BD, therefore each of the figures EBCA, DBCF is a parallelogram; and EBCA is equal o to b. 35. ti DBCF, because they are upon the same base BC, and between the same parallels BC, EF; and the triangle ABC is the half of the pa с rallelogram Book I. rallelogram EBCA, because the diameter AB bisects it; and the triangle DBC is the half of the parallelogram DBCF, because the C. 34, 1. diameter DC bisects it. but the halves of equal things are equal d; d. 7. Ax. therefore the triangle ABC is equal to the triangle DBC. Where fore triangles, &c. Q. E. D. TRIANGLEs upon equal bases, and between the same parallels, are equal to one another. Let the triangles ABC, DEF be upon equal bases BC, EF, and between the same parallels BF, AD. the triangle ABC is equal to the triangle DEF. Produce AD both ways to the points G, H, and thro'B draw BG å. 31. 3. parallel to CA, and thro' F draw FH parallel to ED. then each of the figures GBCA, G А. D H. DEFH is a parallelo gram; and they are 8. 36. 1. equal 6 to one ano ther, because they B CE F €. 34.5, GH; and the triangle ABC is the half of the parallelogram GBCA, because the diameter AB bifects it; and the triangle DEF is the half of the parallelogram DEFH, because the diameter DF bisects d. 7. Ax. it. but the halves of equal things are equal d; therefore the tri angle ABC is equal to the triangle DEF. Wherefore triangles, &c. . E, D. EQUAL triangles upon the same base, and upon the fame side of it, are between the same parallels. Let the equal triangles ABC, DBC be upon the fame base BC, and upon the fame fide of it; they are between the same parallels. Join AD; AD is parallel to BC; for if it is not, thro’ the point 2. 31. 3. A draw : AE parallel to BC, and join EC, the triangle ABC is equat equal 6 to the triangle EBC, because it is upon the fame base BC, Book I. and between the same parallels BC, AE. А D but the triangle ABC is equal to the tri b, 37 11 E С PROP. XL. THEOR. EQUAL triangles upon equal bases, and towards the same parts, are between the same parallels. 2. 31.8. Let the equal triangles ABC, DEF be upon equal bases BC, EF, А D G B СЕ, F B. 38.65 THEOR. PROP. XLI. and between the same parallels; the parallelogram shall са Let |