Page:The Elements of Euclid for the Use of Schools and Colleges - 1872.djvu/36

 PROPOSITION 6. THEOREM. If two angles of a triangle be equal to one another, the sides also which subtend, or are opposite to, the equal angles, shall be equal to one another. Let ABC be a triangle, having the angle ABC equal to the angle ACB: the side AC shall be equal to the side AB. For if AC be not equal to AB, one of them must be greater than the other. Let AB be the greater, and from it cut off DB equal to AC the less, and join DC. Then, because in the triangles DBC, ACB, DB is equal to AC, and BC is common to both, the two sides DB, BC are equal to the two sides AC, CB each to each; and the angle DBC is equal to the angle ACB; therefore the base DC is equal to the base AB, and the triangle DBC is equal to the triangle ACB, the less to the greater; which is absurd. Therefore AB is not unequal to AC, that is, it is equal to it. Wherefore, if two angles &c. Corollary. Hence every equiangular triangle is also equilateral. PROPOSITION 7. THEOREM. On the same base, and on the same side of it, there cannot be two triangles having their sides which are terminated at one extremity of the base equal to one another, and likewise those which are terminated at the other extremity equal to one another. If it be possible, on the same base AB, and on the same side of it, let there be two triangles ACB, ADB, having their sides CA, DA, which are terminated at the extremity A of the base, equal