Let the triangles aei and oiu, be proportionall in shanks: As ae is to ai, so let io be to ou: And let ea bee parallel to io: And ai to ou: Item, let them make the angle aio, betweene them, to wit, betweene their middle shankes ai, and oi, I say their bases ei, and iu, are but one right line continued. For seeing that by the grant ae, and oi, are parallels: Item ai and uo, the right line ai and oi, shall make, by the [21 e v], the angles at a, and o, equall to the alterne angle aio: And therefore they are equall betweene themselves: And then, by the [14 e], the triangles given are equiangles: Therefore the angle oui, is equall to the angle aie: Wherfore the three angles oiu, oia, and aie, by the [3 e], are equall to the three angles of the triangle eai, which are equall by the [13 e vj]. Unto two right angles: And therefore they themselves also are equall to two right angles. Wherefore, by the [14 e v], ei, and iu, are one right line continued.
16. If two triangles have one angle equall, another proportionall in shankes, the third homogeneall, they are equiangles. 7. p. vj.
Let aei, and ouy, the triangles given be equall in their angles a, and o: and proportionall in the shankes of the angles e, and u: and their other angles, at i, and y, homogeneall, that is, let them be both, either acute, or obtuse, or right angles. But first let them be acute, I say, the other at e, & u, are equall. Otherwise let aes, by the [11 e iij.] be made equall to the same ouy; Then have you them by the [4 e], equiangles; and the angles ase, shall be equall to the angle oyu; and both are acute angles: and by the [12. e], aes, and ouy, are proportionall in sides: and as ae, is to es; so shall ou, be to uy, that is, by the grant, so shall ae, be to ei. Therefore because the same ea, hath unto two, to wit, es, and ei, the same reason, the said es, and ei, are equall one to another: And therefore, by the [17. e. vj.] the angles at the base in s and i, are equall. Therefore both of them are acute angles: And in like manner ase, is an acute angle, contrary to the [14. e v]. The same will fall out altogether like to both the other, being either obtuse or right angles. The last part of a right angle is manifest by the [4 e] of this Booke.
Of Geometry the eight Booke, of the diverse kindes of Triangles.
1. A triangle is either right angled, or obliquangled.
The division of a triangle, taken from the angles, out of their common differences, I meane, doth now follow. But here first a speciall division, and that of great moment, as hereafter shall be in quadrangles and prismes.