11. If the base of a triangle doe subtend a right-angle, the rectilineall fitted to it, shall be equall to the like rectilinealls in like manner fitted to the shankes thereof: And contrariwise, out of the 31. p. vj.
Or thus: If the base of a triangle doe subtend a right angle, the right lined figure made upon the base, is equall to the right lined figures like, and in like manner situate upon the feete: H.
Let the right angled triangle be aei: and let there be also the triangles eau, and aiy, and to them upon the base of the said right angle, by the [23 e iiij]. let the triangle ies, be made like, and in like manner situate. I say, that eis, is equall joyntly to eau, and aiy. Let ao, a perpendicular fall from the right angle a, to the base ei: This by the ioe, doth yeeld us twise three proportionals, to wit, ie, ea, eo: Item, ei, ia, io: Therefore, by the [25. e. iiij], as ie, is to eo: so is the triangle ies, to the triangle eau; And as ei, is to oi, so is the triangle eis, to the triangle aiy: But ei, is equall to eo, and oi, the whole, to wit, to his parts. Wherefore by the second composition in
Arithmeticke (9. c. ij.) the triangle eis, is equall to the triangles eau, and iay.
The Converse is thus proved: Let the triangle be aei: And let the perpendicular eo, be erected upon ae, equall to ei: And draw a right line from o to a: Here by the former, the rectilinealls situate at oe, and ea, that is by the construction, at ae, and ie, are equall to the rightilineall at ao, made alike and situate alike: And by the graunt they are equall, to the rectilineall at ai, made alike and situated alike. Therefore seeing the like rectilineals at ao, and ai, are equall; they have by the [20 e iiij], their homologall sides equall: And the two triangles are equiliters: And by the [1 e vij], equiangles. But aeo, is a right angle, by the construction: And aei, is proved to be equall to the same aeo: Therefore, by the [13 e v]. aei, also is a right angle.
12. An obliquangled triangle is either Obtusangled or Acutangled.
The division of an obliquangled triangle is taken from the speciall differences of an oblique angle. For at the 15 e iij, we were taught that an oblique angle was either obtuse or acute: Therefore an obliquangled triangle is an obtuseangle, and an Acutangle.
13. An obtusangle is that triangle which hath one blunt corner. 28.d i.
There can be but one right angle in a triangle, by the [2 e]. Therefore also in it there can be but one blunt angle.