The second and third parts may be concluded out of the first. The second is thus: Twise two angles are equall to two right angles oyu, and euy, by the former part: Item, auy, and euy, by the [14 e]. Therefore they are equall betweene themselves. Now from the equall, Take away euy, the common angle, And the remainders, the alterne angles, at u, and y shall be least equall.
The third is thus: The angles euy, and oys, are equall to the same uyi, by the second propriety, and by the [15 e]. Therefore they are equall betweene themselves.
The converse of the first is here also the more manifest by that light of the common perpendicular, And if any man shall thinke, That although the two inner angles be equall to two right angles, yet the right may meete, as if those equall angles were right angles, as here; it must needes be that two right lines divided by a common perpendicular, should both leane, the one this way, the other that way, or at least one of them, contrary to the [13 e ij].
If they be oblique angles, as here, the lines one slanting or
obliquely crossing one another, the angles on one side will grow lesse, on the other side greater. Therefore they would not be equall to two right angles, against the graunt.
From hence the second and third parts may be concluded. The second is thus: The alterne angles at u and y, are equall to the foresayd inner angles, by the 14 e: Because both of them are equall to the two right angles: And so by the first part the second is concluded.
The third is therefore by the second demonstrated, because the outter oys, is equall to the verticall or opposite angle at the top, by the [15 e]. Therefore seeing the outter and inner opposite are equall, the alterne also are equall.
Wherefore as Parallelismus, parallell-equality argueth a three-fold equality of angles: So the threefold equality of angles doth argue the same parallel-equality.
Therefore,