The converse, is concluded by the same sorites, but by saying all backward. For ua unto ae is, as uao is unto oae, by the 7 e: And as eai, by the grant: Because they are equall: And as ia is unto ao, by the same, Wherefore ua is unto ae, as ia is unto ao.
12. If two triangles be equiangles, they are proportionall in shankes: And contrariwise: 4 and 5. p. vj.
The comparison both of the rate and proportion of triangles hath in the former beene taught: Their similitude remaineth for the last place. Which similitude of theirs consisteth indeed of the reason, or rate of their angles and proportion of the shankes. Therefore for just cause was the reason of the angles set first: Because from thence not onely their reason, but also their latter proportion is gathered. Let aei and iou, be two triangles equiangled: And let them be set upon the same line eiu, meeting or touching one another in the common point i. Then, seeing that the angles at e and i, are granted to bee equall, the lines oi, and ae, are parallel, by the [21 e v]. Therefore by the [22 e v] uo and ea, being continued, shall meete. Item, The right lines ai, and yu, by the [21 e v], are parallel, because the angle aie is equall to oui, the inner opposite to it. Therefore seeing that ai is parallell to the base yu, by the [21 e v], ea shall be to ay, that is, by the [26 e v], to io, as ei is to iu: And alternly, or crosse wayes, ea shall be to ei, as io is to iu. This is the first proportion. Item,
seeing that io is parallell to the base ye; yo, that is, by the [26 e v], ai shall bee unto ou, as ei, is unto iu: And crosse wise, as ai is unto ie, so is ou unto ui. This is the second proposition. Lastly, equiordinately: ae is to ai, as oi is to ou: wherefore if triangles be equiangled, they are proportional in shankes.
This converse is thus demonstrated. Let there be two triangles aei, and ouy, proportionall in shankes: And as ae is to ei; so let ou, be to uy: And as ai is to ie; so let oy bee to yu. Then at the points u and y, let angles be made by the [11 e iij]. equall to the angles at e and i, and let the triangle uys, be made: for the other angles at a and s, shall be equall by the [4 e]. And the triangle yus, shall be equiangled to the assigned aei. And by the antecedent, it shall be proportionall to it in shankes. Thus are two triangles ouy, by the grant; and uys, by the construction, proportionall in shanks to the same triangle aei: And as ae, is to ei, so is ou, to uy; so is su, to uy. Therefore seeing ou and su, are proportionall to the same yu, they are equall; Item, as ai is to ie: so is oy unto yu: so also is sy unto yu. Therefore oy and sy, seeing they are proportionall to the same yu, are equall. (yu is the common side.) The triangle therefore ouy, is equilater unto the triangle syu. And by the [1 e], it is to it equiangle: And therefore it is equiangled to the triangle aei, which was to be prooved. This was generally before taught at the [20 e iiij], of homologall sides subtending equall angles.
Therefore,
13. If a right line in a triangle be parallell to the base, it doth cut off from it a triangle equiangle to the whole, but lesse in base.