Page:Euclid's Elements 1714 Barrow translation.djvu/23

Rh and AC = DF) and have the angle A equal to the angle D contained under the equal right lines; they hall have the bae EC equal to the bae EF; and  the triangle BAC hall be equal to the triangle EDF; and the remaining angles B, C, hall be equal to the remaining angles E, F, each to each, under which the equal ides are ubtended.

If the point D be applied to the point A, and the right line DE plac'd upon the right line AB, the point E hall fall upon B,becaue DE a = AB, alo the right line DF hall fall upon AC,becaue the angle A a = D. moreover the point F hall fall upon the point C, becaue AC a = DF. Therefore the right lines EF, BC hall agree, becaue they have the ame Terms, and conequently are equal. Wherefore the triangles BAC, DEF, and the angles B, E, as alo the angles C, F, do agree, and are equal. Which was to be Demontrated.

The angles ABC, ACB, at the bae of an Ioceles triangle ABC, are equalone to the other: And if the equal ides AB, AC be produc'd, the angles CBD, BCE, under the bae, hall be equal one to the other.

a Take AE= AD; and b join CD, and BE.

Becaue, in the triangles ACD, ABE, are AB c = AC, and AE d = AD, and the angle A common to them both, e therefore is the angle ABE = ACD,and the angle AEB e = ADC, and the bae BE e = CD; alo EC f = DB. Therefore in the triangles BEC, BDC g hall be the angle ECB = DBC. Which was to be Dem. Alo therefore the angle EBC = DCB. but the angle ABE h = ACD; therefore the angle ABC k = ACB. Which was to be Dem.