As in the triangle aei, the right line ou, doth cut off the triangle aou, equiangle, by the [21 e v], to the whole aei; But the base ou, is lesse than the base ei, as appeareth by the [21 e], and by the alternation of the sides.
14. If two trangles be proportionall in the shankes of the equall angle, they are equiangles: 6 p vj.
Let therefore the triangles given be aei, and ouy, equall in their angles a and o: And in their shankes let ea, be unto ai, as ou is to oy: And by the [11 e iij], let the angles soy, and oys, be equall to the angles eai, and eia: The other at s and e, shall be equall, by the [4 e]. Here thou seest that the triangle aei, is equiangle unto oys. Now, by the [12 e.] as ea is to ai: so is so to oy: and therefore, by the grant, so is uo to oy. Therefore seeing that uo, and os, are proportionall to oy, they are both equall. Lastly, if the common shanke oy bee added to both the shankes ou, and oy, are equall to the shankes so and oy. [But by the construction the angles oys and aie are equall. And, by the [4 e], the other at s and e are equall. Therefore the first triangle aei, is made equiangled to the third. Now seeing the second triangle uoy is to the third soy, equall in the shanks of the equall angle, it is to the same equilater, and by the [1 e], equiangled: Shon.] Wherefore the second triangle ouy shall likewise be equiangled to osy, the third: And therefore if
two triangles proportionall in shankes be equall in the angle of their shankes, they are equiangles.
15. If triangles proportionall in shankes, and alternly parallell, doe make an angle betweene them, their bases are but one right line continued. 32 p. vj.
Or thus: If being proportionall in their feet, and alternately parallels, they make an angle in the midst betweene them, they have their bases continued in a right line: H.
The cause is out of the [14 e v]. For they shall make on each side, with the falling line ai, two angles equall to two right angles.