ere
as sixe triangles made betwene those two gemowe lines A.B. and
C.F, the first triangle is A.C.D, the seconde is A.D.E, the
thirde is A.D.B, the fourth is A.B.E, the fifte is D.E.B, and
the sixte is B.E.F, of which sixe triangles, A.D.E. and D.E.B.
are equall, bicause they haue one common grounde line. And so
likewise A.B.E. and A.B.D, whose commen grounde line is A.B, but
A.C.D. is equal to B.E.F, being both betwene one couple of
parallels, not bicause thei haue one ground line, but bicause
they haue their ground lines equall, for C.D. is equall to E.F,
as you may declare thus. C.D, is equall to A.B. (by the foure
and twenty Theoreme) for thei are two contrary sides of one
lykeiamme. A.C.D.B, and E.F by the same theoreme, is equall to
A.B, for thei ar the two y^e contrary sides of the likeiamme,
A.E.F.B, wherfore C.D. must needes be equall to E.F. like wise
the triangle A.C.D, is equal to A.B.E, bicause they ar made
betwene one paire of parallels and haue their groundlines like,
I meane C.D. and A.B. Againe A.D.E, is equal to eche of them
both, for his ground line D.E, is equall to A.B, inso muche as
they are the contrary sides of one likeiamme, that is the long
square A.B.D.E. And thus may you proue the equalnes of all the
reste.

_The xxix. Theoreme._

Al equal triangles that are made on one grounde line, and
rise one waye, must needes be betwene one paire of
parallels.

_Example._

Take for example A.D.E, and D.E.B, which (as the xxvij.
conclusion dooth proue) are equall togither, and as you see,
they haue one ground line D.E. And againe they rise towarde one
side, that is to say, vpwarde toward the line A.B, wherfore they
must needes be inclosed betweene one paire of parallels, which
are heere in this example A.B. and D.E.

_The thirty Theoreme._

Equal triangles that haue their ground lines equal, and be
drawen toward one side, ar made betwene one paire of
paralleles.

_Example._

The example that declared the last theoreme, maye well serue to
the declaracion of this also. For those ij. theoremes do diffre
but in this one pointe, that the laste theoreme meaneth of
triangles, that haue one ground line common to them both, and
this theoreme dothe presuppose the grounde lines to bee diuers,
but yet of one length, as A.C.D, and B.E.F, as they are ij.
equall triangles approued, by the eighte and twentye Theorem, so
in the same Theorem it is declared, y^t their ground lines are
equall