to each, namely, ABC to DEF, and BCA to EFD; and let them have also one side equal to one side; and first let those sides be equal which are adjacent to the equal angles in the two triangles, namely, BC to EF: the other sides shall be equal, each to each, namely, AB to DE, and AC to DF, and the third angle BAC equal to the third angle EDF.
![float](http://upload.wikimedia.org/wikipedia/commons/thumb/e/eb/The_Elements_of_Euclid_for_the_Use_of_Schools_and_Colleges_-_1872_page_29.png/320px-The_Elements_of_Euclid_for_the_Use_of_Schools_and_Colleges_-_1872_page_29.png)
For if AB be not equal to DE, one of them must be greater than the other. Let AB be the greater, and make BG equal to DE, [I. 3.
and join GC.
Then because GB is equal to DE, [Construction.
and BC to EF; [Hypothesis.
the two sides GB, BC are equal to the two sides DE, EF, each to each;
and the angle GBC is equal to the angle DEF; [Hypothesis.
therefore the triangle GBC is equal to the triangle DEF, and the other angles to the other angles, each to each, to which the equal sides are opposite; [I. 4.
therefore the angle GCB is equal to the angle DFE.
But the angle DFE is equal to the angle ACB. [Hypothesis.
Therefore the angle GCB is equal to the angle ACB, [Ax. 1.
the less to the greater; which is impossible.
Therefore AB is not unequal to DE,
that is, it is equal to it;
and BC is equal to EF; [Hypothesis.
therefore the two sides AB, BC are equal to the two sides DE, EF, each to each;
and the angle ABC is equal to the angle DEF; [Hypothesis.
therefore the base AC is equal to the base DF, and the third angle BAC to the third angle EDF. [I. 4.