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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.
AB = DE,
..J AB concides with DE, Hyp
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.
PROP. V. THEOR.
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