adjacent to thofe fides in the a QRS = to the <s correfponding to them in the Triangle qrs, all the reft, and alfo the Trianglès themfelves, will be equal. For if the fide q r be put upon the fide QR, they will agree (by Maxim 7.) but b.eaufe the <R = <r, and < S = <s; the fide Rq will fall upon fide RQ, and qs upon QS,.. the point q will fall upon the pointQ; (for if it fall out of 2, the Lines r q, qs do not fall upon the Lines QR, 25) therefore they are equal (by the feventh Maxim) QED. THEOREM II. Fig.37. In an Ifoceles Triangle the Angle at the Base oppofite to the equal Legs, are equal. Let the Triangle A CB be fuppofed two Triangles, and the Situation of one con vers to that of the other, as bca: Becaufe in the two Triangles ACB and bca, the fide AC is the fide b c, and the fide CB is the fide ab, and the Angle C is the Angle c, therefore the Angle at the Base A =< b by the first, which was to be Demonftrated for the Angle, B and b are the fame.. THEO THEOREM III. Fig. 38. If two Triangles have each fide in one equal to its correfponding fide in the other (that is, ac = ef, cb=fi, and ab = ei) they will alfo have = Angles oppofite to thofe fides (that is c = f, a=e,and b=i). = Let the line ab be put upon the line ei. Then the point c will either fall in f, or it will not. If it falls in f, the whole Triangles agree, and therefore all the Angles are equal by the feventh Maxim. Fig 39. If c falls out of f, draw the line fc Becaufe by Hypothefis the fides ef, and a c are equal, the Angle efc muft be equal to the Angle ecf; by the fecond Propofition therefore Angleife will be greater than Angle ecf, and Angle ifc wil be much greater than Angle icf. Again, by fuppofition, becaufe if= be, <ife, will be to <icf. Therefore ifcis both much greater than, and equal to cf, which is impoffible, and therefore a cannot fall out of f. THEOREM IV. = Fig.40.One right line CDfalling on another AB, makes the Contiguous Angles to two right Angles. Let the line CD be perpendicular to AB, then the Angles ADC and CDB : will be right pr.5 Def. But if the line fall obliquely as ED, raife the perpendicular CD, then, because the unequal Angles ADE and EDB occupy the fame place which the two right ones ADC and CDB did, therefore they agree to two right Angles, and confequently to them (by 7 Ax.) Q. E.D. THEOREM V. Fig.41 If two right lines (BC,and FL) cut one the other in any point (A) the oppofite Angles at the Vertex (A) will be equal; that is, the Angle LAB is to CAF. Because BA, ftands upon the right line LF, LAB, and FAB will be equal to two right (by the fourth Theorem.) And becaufe FA ftands upon BC, the Angles FAC, and FAB, will be equal to two right (by the fame) therefore the two Angles LAB, and FAB taken together will be mal to the two Angles CAF, and FAB, en together. But the common Angle FAB being taken away, there remains the Angle LAB = CAF (by the third Max.) Q.E.D. THEOREM. VI. Fig 42. If a right line GO cut the two parallel right lines (AB CF,) first the alternate <(RLO, QOL, and BLO,COL.) are equals. Secondly, The external Angle (GLB) is equal. to to the internal one (LOF,) as alfo(< GLR= <LOC.) Thirdly, The two internal Angles; towards the fame parts together ALO COL= to two right Angles: Alfo the <s BLO, FOL together to two right Angles. Demon. of the first part Draw LQ and RO perpendicular toCF from the points LandO and they will be alfo perpendicular to AB by the 12th Maxim,) now in the as ROL, LOQ, the fide OL, is common to both; RO=LQ (by the 8th Def.)and ...<LOR=<LOQ (by the first Prop.) alfo <ROL=<QLO.. < BLO COL (by the fecond Max.) which is the First Part. Second Part. Angle LOQ =<ROL (by the Firft Part,) and <RLO= <GLB (by the 5th Prop.)LOQ=<GLB (by the firft Max.) after the fame manner may <LOC be proved tc< GLR which is the fecond Part. The 3d. Part. Angle GLB =<RLO (by the 5th Prop.) <COL was proved = < RLG (in the 2d. Part.) but◄ RLG +< GLB = two right <s (by the 4th Prop.) :.< COL <RLO = two right Angles, as alfo < BLO +- < FOL Q. E. D. THEOREM VII. In any Triangle abc, the three Angles taken together, are equal to two right ones. |