The Path-Way to Knowledg, Containing the First Principles of Geometrie Part 14

Whan so euer in any triangle the line of one side is drawen forthe in lengthe, that vtter angle is greater than any of the two inner corners, that ioyne not with it.

_Example._

[Ill.u.s.tration]

The triangle A.D.C hathe hys grounde lyne A.C. drawen forthe in lengthe vnto B, so that the vtter corner that it maketh at C, is greater then any of the two inner corners that lye againste it, and ioyne not wyth it, whyche are A. and D, for they both are lesser then a ryght angle, and be sharpe angles, but C. is a blonte angle, and therfore greater then a ryght angle.

_The tenth Theoreme._

In euery triangle any .ij. corners, how so euer you take th?, ar lesse th? ij. right corners.

_Example._

[Ill.u.s.tration]

In the firste triangle E, whiche is a threlyke, and therfore hath all his angles sharpe, take anie twoo corners that you will, and you shall perceiue that they be lesser then ij. right corners, for in euery triangle that hath all sharpe corners (as you see it to be in this example) euery corner is lesse then a right corner. And therfore also euery two corners must nedes be lesse then two right corners. Furthermore in that other triangle marked with M, whiche hath .ij. sharpe corners and one right, any .ij. of them also are lesse then two right angles. For though you take the right corner for one, yet the other whiche is a sharpe corner, is lesse then a right corner. And so it is true in all kindes of triangles, as you maie perceiue more plainly by the .xxij. Theoreme.

_The .xi. Theoreme._

In euery triangle, the greattest side lieth against the greattest angle.

_Example._

[Ill.u.s.tration]

As in this triangle A.B.C, the greattest angle is C. And A.B.

(whiche is the side that lieth against it) is the greatest and longest side. And contrary waies, as A.C. is the shortest side, so B. (whiche is the angle liyng against it) is the smallest and sharpest angle, for this doth folow also, that is the longest side lyeth against the greatest angle, so it that foloweth

_The twelft Theoreme._

In euery triangle the greattest angle lieth against the longest side.

For these ij. theoremes are one in truthe.

_The thirtenth theoreme._

In euerie triangle anie ij. sides togither how so euer you take them, are longer th? the thirde.

[Ill.u.s.tration]

For example you shal take this triangle A.B.C. which hath a very blunt corner, and therfore one of his sides greater a good deale then any of the other, and yet the ij. lesser sides togither ar greater then it. And if it bee so in a blunte angeled triangle, it must nedes be true in all other, for there is no other kinde of triangles that hathe the one side so greate aboue the other sids, as thei y^t haue blunt corners.

_The fourtenth theoreme._

If there be drawen from the endes of anie side of a triangle .ij. lines metinge within the triangle, those two lines shall be lesse then the other twoo sides of the triangle, but yet the corner that thei make, shall bee greater then that corner of the triangle, whiche standeth ouer it.

_Example._

[Ill.u.s.tration]

A.B.C. is a triangle. on whose ground line A.B. there is drawen ij. lines, from the ij. endes of it, I say from A. and B, and they meete within the triangle in the pointe D, wherfore I say, that as those two lynes A.D. and B.D, are lesser then A.C. and B.C, so the angle D, is greatter then the angle C, which is the angle against it.

_The fiftenth Theoreme._

If a triangle haue two sides equall to the two sides of an other triangle, but yet the gle that is contained betwene those sides, greater then the like angle in the other triangle, then is his grounde line greater then the grounde line of the other triangle.

[Ill.u.s.tration]

_Example._

A.B.C. is a triangle, whose sides A.C. and B.C, are equall to E.D. and D.F, the two sides of the triangle D.E.F, but bicause the angle in D, is greatter then the angle C. (whiche are the ij. angles contayned betwene the equal lynes) therfore muste the ground line E.F. nedes bee greatter thenne the grounde line A.B, as you se plainely.

[Ill.u.s.tration]

_The xvi. Theoreme._

If a triangle haue twoo sides equalle to the two sides of an other triangle, but yet hathe a longer ground line th? that other triangle, then is his angle that lieth betwene the equall sides, greater th? the like corner in the other triangle.

_Example._

This Theoreme is nothing els, but the sentence of the last Theoreme turned backward, and therfore nedeth none other profe nother declaration, then the other example.

_The seuententh Theoreme._

If two triangles be such sort, that two angles of the one be equal to ij. angles of the other, and that one side of the one be equal to on side of the other, whether that side do adioyne to one of the equall corners, or els lye againste one of them, then shall the other twoo sides of those triangles bee equalle togither, and the thirde corner also shall be equall in those two triangles.

_Example._

[Ill.u.s.tration]

Bicause that A.B.C, the one triangle hath two corners A. and B, equal to D.E, that are twoo corners of the other triangle.

D.E.F. and that they haue one side in theym bothe equall, that is A.B, which is equall to D.E, therefore shall both the other ij. sides be equall one to an other, as A.C. and B.C. equall to D.F. and E.F, and also the thirde angle in them both shal be equall, that is, the angle C. shal be equall to the angle F.

_The eightenth Theoreme._

When on ij. right lines ther is drawen a third right line crosse waies, and maketh .ij. matche corners of the one line equall to the like twoo matche corners of the other line, then ar those two lines gemmow lines, or paralleles.

_Example._