The elements of geometrie of the most auncient philosopher Euclide of Megara. Faithfully (now first) translated into the Englishe toung, by H. Billingsley, citizen of London. Whereunto are annexed certaine scholies, annotations, and inuentions, of the best mathematiciens, both of time past, and in this our age. With a very fruitfull præface made by M. I. Dee, specifying the chiefe mathematicall scie[n]ces, what they are, and wherunto commodious: where, also, are disclosed certaine new secrets mathematicall and mechanicall, vntill these our daies, greatly missed

The 7. Theoreme. The 7. Proposition. If there be two triāgles, wherof the one hath one angle equal to one angle of the other, and the sides which include the other angles, be proportionall, and if either of the other angles re∣mayning be either lesse or not lesse then a right angle: thē shal the triangles be equiangle, and those angles in them shall be equall, which are contayned vnder the sides proportionall.

SVppose that there be two triangles ABC and DEF, which let haue one angle of the one, equall to one angle of the other, namely, the angle BAC equall vnto the angle EDF. And let the sides which include the other angles, namely, the angles ABC and DEF be proportio∣nall, so that as AB is to BC, so let DE be to EF. And let the other angles re∣mayning, namely, ACD and DFE be first either of them lesse then a right angle. Then I say that the triangle ABC is equiangle vnto the triangle DEF. And that the angle ABC is equall vnto the an∣gle

[illustration]

DEF,* 1.1 namely, the angles which are contai∣ned vnder the sides proportionall, and that the angle remayning, namely, y^e angle C is equall vn∣to the angle remayning, namely, to y^e angle F. For first the angle ABC is either equall to the an∣gle DEF, or els vnequall. If the angle ABC be equall to the angle DEF,* 1.2 then the angle remai∣ning, namely, ACB, shall be equall to the angle remayning DFE (by the corollary of the 32. of the first) And therfore the triangles ABC and DEF are equiangle. But if the angle ABC be vnequall vnto the angle DEF, then is the one of them grea¦ter then the other. Let the angle ABC be the greater, and vnto the right line AB and vnto the point in it B (by the 23. of the first) describe vnto the angle DEF an equall angle ABG. And forasmuch as the angle A is equall vnto the angle D, and the angle ABG is equall vnto the angle DEF, therfore the an∣gle remayning AGB is equall vnto the angle remayning DFE (by the corol∣lary of the 32. of the first. Wherfore the triangle ABG is equiangle vnto the tri∣angle DEF. Wherfore (by the 4. of the sixth) as the side AB is to the side BG, so is the side DE to the side EF. But by suppposition the side DE is to the side EF, as the side AB is to the side BC. Wherfore (by the 11. of the fifth) as the side AB is to the side BC so is the same side AB to the side BG. Wher∣fore AB hath to either of these BC and BG one and the same proportion, and therfore (by the 9. of the fifth) BC is equall vnto BG. Wherefore (by the 5. of the first) y^e angle BGC is equall vnto y^e angle BCG: but by supposition y^e angle BCG is lesse then a right angle. Wherfore the angle BGC is also lesse then a right angle. Wherfore (by the 13. of the first) the side angle vnto it, namely, AGB is greater then a right angle, and it is already proued that the same angle is equall vnto the angle F. Wherfore the angle F is also greater then a right angle. But it is supposed to be lesse which is absurde. Wherefore the angle ABC is not vnequall vnto the angle DEF, wherfore it is equall vnto it. And the angle A is equall vnto the angle D by supposition. Wherfore the angle remayning, name∣ly, C, is equall vnto the angle remayning, namely, to F (by the corollary of the 32. of the first) Wherfore the triangle ABC is equiangle vnto the triangle DEF.* 1.3

But now suppose that either of the angles ACB and DFE be not lesse then

a right angle. That is, let either of them be a right angle, or either of them grea∣ter then a right angle. Then I say againe that in that case also the triangle ABC is equiangle vnto the triangle DEF. For if either of them be a right angle, forasmuch as all right angles are (by the 4. peticion) equall the one to the other, straight way will follow the intent of the proposition. But if either of them be greater then a right angle, then the same order of construction that was before being kept, we may in like sort proue that the side BC is equall vnto the side BG. Wherfore also the angle BCG is equall vnto the angle BGC. But the angle BCG is greater then a right angle. Wherfore also the angle BGC is greater thē a right angle. Wherfore two angles of the triangle BGC are greater then two right angles: which (by the 17. of the first) is impossible. Wherfore the angle ABC is not vnequall vnto the angle DEF. And therfore is it equall: but the angle A is equall vnto the angle D (by supposition) Wher∣fore

[illustration]

the angle remayning, namely, C is equal vnto the angle remayning, namely, to F (by the corollary of the 32. of the first). Wherfore the triangle ABC is equi∣angle vnto the triangle DEF. If there••ore there be two triangles whero•• the one hath one angle equall to one angle o•• the other, and the sides which include the other angle be proportionall, and if either of the other angles remaining be either lesse or not lesse then a right angle, the triangle shall be equiangle, and those angles in them shall be equall which are contained vnder sides proportionall: which was required to be proued.

Notes

• * 1.1

  The first part of this proposition.
• * 1.2

  Demonstra∣tion leading to an impossibi∣litie.
• * 1.3

  The second part of this proposition.