As here in the triangle aei, let ou, be parallell to the base; and let a third parallel be understood to be in the toppe a; therefore, by the [28. e. v]. the intersegments are proportionall.
The converse is forced out of
the antecedent: because otherwise the whole should be lesse than the part. For if ou, be not parallell to the base ei, then yu, is: Here by the grant, and by the antecedent, seeing ao, oe, ay, ye, are proportionall: and the first ao, is lesser than ay, the third: oe, the second must be lesser than ye, the fourth, that is the whole then the part.
13. The three angles of a triangle, are equall to two right angles. 32. p j.
Hitherto therefore is declared the comparison in the sides of a triangle. Now is declared the reason or rate in the angles, which joyntly taken are equall to two right angles.
The truth of this proposition, saith Proclus, according to common notions, appeareth by two perpendiculars erected upon the ends of the base: for looke how much by the leaning of the inclination, is taken from two right angles at the base, so much is assumed or taken in at the toppe, and so by that requitall the equality of two right angles is made; as in the triangle aei, let, by the [24. e v], ou, be parallell against ie. Here three particular angles, iao, iae, eau, are equall to two right lines; by the [14. e v]. But the inner angles are equall to the same three: For first, eai, is equall to it selfe: Then the other two are equall to their alterne angles, by the [24. e v].
Therefore
14. Any two angles of a triangle are lesse than two right angles.