. Elements of geometry : containing books I to III. hen the following partsare equal in the two triangles. 1. Two sides and the angle between them. I. 4. 2. Two angles and the side between them. I. 26. 3. The three sides of each. I. 8. 4. Two angles and the side opposite one of them. I. 26. The Propositions, in which these cases are proved, are themost important in our First Section. The first case we have proved in Prop. iv. Availing ourselves of the method of superposition, we canprove Cases 2 and 3 by a process more simple than that em-ployed by Euclid, and with the further advantage of bri
. Elements of geometry : containing books I to III. hen the following partsare equal in the two triangles. 1. Two sides and the angle between them. I. 4. 2. Two angles and the side between them. I. 26. 3. The three sides of each. I. 8. 4. Two angles and the side opposite one of them. I. 26. The Propositions, in which these cases are proved, are themost important in our First Section. The first case we have proved in Prop. iv. Availing ourselves of the method of superposition, we canprove Cases 2 and 3 by a process more simple than that em-ployed by Euclid, and with the further advantage of bringingthem into closer cnnexion with Case 1. We shall thereforefp% ? three Propositions, which we <!• signate A, B, and C, inthe Place of Euclids Props, v. vi. vn. vm. The displaced Propositions will be found on pp. 108-112. Proposition A corresponds with Euclid I. 5. B ... I. 2i, first part C I. 8. lsides must also be In the isosceles triangle ABC, let AC=AB. (Fig. 1.) Then must i ABG= i ACB [magine the a ABC to be taken up, turned round, and setdown again in a reversed position as in Fig. 2, and designatethe angular points A, B, C. Then in As ABC, .(/., AB I •. and AC= A TV. and iBAC=i CAB, :. i A BO= l A CB. I. 4. But lACB^iACi:, :. l ABC= t ACB. ax. 1. [frnrc overy equilateral triangle is also equiangular. Nora. When one side fa triangle is distinguished fromth- her rides by being called the Base, the angular point op-idi i i - - ? i the triangle. Book PKOPOSIT/OX B. 17 Proposition 13. Throrkm. If two triangh - : wd to twoangles of th oikt and Ou sides adjacent ti-the equal angl also equal; then must tht triangles be equal in alt
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