The elements of Euclid for the use of schools and colleges : comprising the first two books and portions of the eleventh and twelfth books; with notes and exercises . de with the base EF, two straight Hues willenclose a space ; which is impossible. {Axiom 10. Therefore the base BG coincides with the base EF, and isequal to it. {Axiom 8. Therefore the whole triangle ABG coincides with the wholetriangle DEF, and is equal to it. {Axiom 8. And the other angles of the one coincide with the otherangles of the other, and are equal to them, namely, theangle ABG to the angle DEF, and the angle AGB to t
The elements of Euclid for the use of schools and colleges : comprising the first two books and portions of the eleventh and twelfth books; with notes and exercises . de with the base EF, two straight Hues willenclose a space ; which is impossible. {Axiom 10. Therefore the base BG coincides with the base EF, and isequal to it. {Axiom 8. Therefore the whole triangle ABG coincides with the wholetriangle DEF, and is equal to it. {Axiom 8. And the other angles of the one coincide with the otherangles of the other, and are equal to them, namely, theangle ABG to the angle DEF, and the angle AGB to theangle DFE. Wherefore, iftico triangles &c. PROPOSITION 5. THEOREM. The angles at the base of an isosceles triangle are equalto one another; and if the equal sides he produced theangles on the other side of the hose shall he equal to oneanother. Let ABG be an isosceles triangle, having the side ABequal to the side AG, and let the straight lines AB, AGbe produced to D and E: the angle ABC shall be equal tothe angle AGB, and the angle GBD to the angle BCE. In BD take any point F,and from AEthe greater cuioSAG equal less, [ BOOK I. 5. II. andjoini^C, GB. Because^i^isequalto-46r, \ AB Ui AG^ [Eijpotkesis. the two sides FA, AC dire equal to thetwo sides GA, AB, each to each; andthey contain the angle FA G commonto the two triangles AFC, AGB;therefore the base FC is equal to thebase GB, and the triangle AFC tothe triangle A GB, and the remainingangles of the one to the remainingangles of the other, each to each, towhich the equal sides are opposite, the angle ACF to the angle ABG, and the angleAFC to the angle A GB. [I. 4. And because the whole ^i^ is equal to the whole AG,of which the parts AB, AC are equal, [Hypotlusi^. the remainder ^i^is equal to the remainder CG. [Axiom FC was shewn to be equal to GB ;therefore the two sides BF, FC are equal to the two sidesCG, GB, each to each; and the anglei5i^C was shewn to be equal to the angle CG
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