Part 18

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.

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.

The .ij. fyrst lynes are A.B. and C.D, the thyrd lyne that crosseth them is E.F. And bycause that E.F. maketh ij. matche angles with A.B, equall to .ij. other lyke matche angles on C.D, (that is to say E.G, equall to K.F, and M.N. equall also to H.L.) therfore are those ij. lynes A.B. and C.D. gemow lynes, vnderstand here by lyke matche corners, those that go one way as doth E.G, and K.F, lyke ways N.M, and H.L, for as E.G. and H.L, other N.M. and K.F. go not one waie, so be not they lyke match corners.

The nyntenth Theoreme.

When on two right lines there is drawen a thirde right line crossewaies, and maketh the ij. ouer corners towarde one hande equall togither, then ar those .ij. lines paralleles. And in like maner if two inner corners toward one hande, be equall to .ii. right angles.

Example.

As the Theoreme dothe speake of .ij. ouer angles, so muste you vnderstande also of .ij. nether angles, for the iudgement is lyke in bothe. Take for example the figure of the last theoreme, where A.B, and C.D, be called paralleles also, bicause E. and K, (whiche are .ij. ouer corners) are equall, and lykewaies L. and M. And so are in lyke maner the nether corners N. and H, and G. and F. Nowe to the seconde parte of the theoreme, those .ij. lynes A.B. and C.D, shall be called paralleles, because the ij. inner corners. As for example those two that bee toward the right hande (that is G. and L.) are equall (by the fyrst parte of this nyntenth theoreme) therfore muste G. and L. be equall to two ryght angles.

The xx. Theoreme.

When a right line is drawen crosse ouer .ij. right gemow lines, it maketh .ij. matche corners of the one line, equall to two matche corners of the other line, and also bothe ouer corners of one hande equall togither, and bothe nether corners like waies, and more ouer two inner corners, and two vtter corners also towarde one hande, equall to two right angles.

Example.

Bycause A.B. and C.D, (in the laste figure) are paralleles, therefore the two matche corners of the one lyne, as E.G. be equall vnto the .ij. matche corners of the other line, that is K.F, and lykewaies M.N, equall to H.L. And also E. and K. bothe ouer corners of the lefte hande equall togyther, and so are M. and L, the two ouer corners on the ryghte hande, in lyke maner N. and H, the two nether corners on the lefte hande, equall eche to other, and G. and F. the two nether angles on the right hande equall togither.

The xxi. Theoreme.

What so euer lines be paralleles to any other line, those same be paralleles togither.

Example.

A.B. is a gemow line, or a parallele vnto C.D. And E.F, lykewaies is a parallele vnto C.D. Wherfore it foloweth, that A.B. must nedes bee a parallele vnto E.F.

The .xxij. theoreme.

In euery triangle, when any side is drawen forth in length, the vtter angle is equall to the ij. inner angles that lie againste it. And all iij. inner angles of any triangle are equall to ij. right angles.

Example.

The triangle beeyng A.D.E. and the syde A.E. drawen foorthe vnto B, there is made an vtter corner, whiche is C, and this vtter corner C, is equall to bother the inner corners that lye agaynst it, whyche are A. and D. And all thre inner corners, that is to say, A.D. and E, are equall to two ryght corners, whereof it foloweth, that all the three corners of any one triangle are equall to all the three corners of euerye other triangle. For what so euer thynges are equalle to anny one thyrde thynge, those same are equalle togitther, by the fyrste common sentence, so that bycause all the .iij. angles of euery triangle are equall to two ryghte angles, and all ryghte angles bee equall togyther (by the fourth request) therfore must it nedes folow, that all the thre corners of euery triangle (accomptyng them togyther) are equall to iij. corners of any triangle, taken all togyther.

The .xxiii. theoreme.

When any ij. right lines doth touche and couple .ij. other righte lines, whiche are equall in length and paralleles, and if those .ij. lines bee drawen towarde one hande, then are thei also equall together, and paralleles.

Example.

A.B. and C.D. are ij. ryght lynes and paralleles and equall in length, and they ar touched and ioyned togither by ij. other lynes A.C. and B.D, this beyng so, and A.C. and B.D. beyng drawen towarde one syde (that is to saye, bothe towarde the lefte hande) therefore are A.C. and B.D. bothe equall and also paralleles.

The .xxiiij. theoreme.