Anguli ad verticem, Angles at the top or head, are called Verticall angles which have their toppes meeting in the same point. The Demonstration is: Because the lines cutting one another, are either perpendiculars, and then all
right angles are equall as heere: Or else they are oblique, and then also are the verticalls equall, as are aui, and oue: And againe, auo, and iue. Now aui, and oue, are equall, because by the [14. e.] with auo, the common angle, they are equall to two right angles: And therefore they are equall betweene themselves. Wherefore auo, the said common angle beeing taken away, they are equall one to another.
And
16. If two right lines cut with one right line, doe make the inner angles on the same side greater then two right angles, those on the other side against them shall be lesser then two right angles.
As here, if auy, and uyi, bee greater then two right angles euy, and uyo, shall bee lesser then two right angles.
17. If from a point assigned of an infinite right line given, two equall parts be on each side cut off: and then from the points of those sections two equall circles doe meete, a right line drawne from their meeting unto the point assigned, shall bee perpendicular unto the line given. 11. p j.
As let a, be the point assigned of the infinite line given: and from that on each side, by the [7. e.] cut off equall
portions ae, and ai, Then let two equall peripheries from the points e, and i, meete, as in o, I say that a right line drawne from o, the point of the meeting of the peripheries. unto a. the point given, shalbe perpendicular upon the line given. For drawing the right lines oe, & oi, the two angles eao, and iao, on each side, equicrurall by the construction of equall segments on each side, and oa, the common side, are equall in base by the [9. e]. And therefore the angles themselves shall be equall, by the [7. e iij]. and therefore againe, seeing that ao, doth lie equall betweene the parts ea, and ia, it is by the [13. e ij]. perpendicular upon it.