- From the greater of two given straight lines, to cut off a part equal to the less. PROP. IV. THEOR. 4. 1 Eu. If two triangles have two sides of the one, equal to two sides of the other, each to each ; and have likewise the angles contained by those sides equal to one another; they shall likewise have their bases, or third sides, equal, and the two triangles shall be equal ; and their other angles shall be equal, each to each, viz., those to which the equal sides are opposite. Let ABC, DEF, be two As, of which AB =DE, AC = DF, and < BAC = LEDF. Then BC = EF, A ABC = A DEF, L ABC = Z DEF, and < ACB = 2 DFE. BCE AB = DE, ..J AB concides with DE, Hyp BAC LEDF; AC falls upon DF. C coincides with F. B coincides with E, For if BC do not coincide with EF, then two Ax. 10. str. lines enclose a space, which is impossible. BC coincides with and = EF, JA ABC coincides with and =A DEF, " LABC coincides with and = _DEF, LACB coincides with and = LDFE. Therefore, if iwo triangles have, &c. PROP. V. THEOR. 5. 1 Eu. are equal to one another; and if the equal :. Therefore, the angles, &c. Cor.-Hence every equilateral A equiangular. is also PROP. V. THEOR. OTHERWISE DEMONSTRATED. are equal to one another. в р с and AB = AC, A ABD = A ACD, 12 ABO = at the base. Note. It is evident that some line, as AD, will bisect the < BAC; and although the method of bisection is not known until Pro. 8 be solved, yet this does not affect the truth of |