Since [L]_A'_ = [L]_B'_, Given and [L]_A_ = [L]_A'_, Hyp. [therefore][L]_A_ = [L]_B'_. [therefore]_B'C'_ will lie along _AC_. Similarly, _A'C'_ will lie along _BC_. Therefore _C'_ will fall on both _AC_ and _BC_, and hence at their intersection. [therefore]_B'C'_ = _AC_. But _B'C'_ was made equal to _BC_. [therefore]_AC_ = _BC_. Q.E.D. If the proposition should be postponed until after the one on the sum of the angles of a triangle, the proof would be simpler, but it is advantageous to couple it with its immediate predecessor. This simpler proof consists in bisecting the vertical angle, and then proving the two triangles congruent. Among the other proofs is that of the _reductio ad absurdum_, which the student might now meet, but which may better be postponed. The phrase _reductio ad absurdum_ seems likely to continue in spite of the efforts to find another one that is simpler. Such a proof is also called an indirect proof, but this term is not altogether satisfactory. Probably both names should be used, the Latin to explain the nature of the English. The Latin name is merely a translation of one of several Greek names used by Aristotle, a second being in English "proof by the impossible," and a third being "proof leading to the impossible." If teachers desire to introduce this form of proof here, it must be borne in mind that only one supposition can be made if such a proof is to be valid, for if two are made, then an absurd conclusion simply shows that either or both must be false, but we do not know which is false, or if only one is false. THEOREM. _Two triangles are congruent if the three sides of the one are equal respectively to the three sides of the other._ It would be desirable to place this after the fourth proposition mentioned in this list if it could be done, so as to get the triangles in a group, but we need the fourth one for proving this, so that the arrangement cannot be made, at least with this method of proof. This proposition is a "partial converse" of the second proposition in this list; for if the triangles are _ABC_ and _A'B'C'_, with sides _a_, _b_, _c_ and _a'_, _b'_, _c'_, then the second proposition asserts that if _b_ = _b'_, _c_ = _c'_, and [L]_A_ = [L]_A'_, then _a_ = _a'_ and the triangles are congruent, while this proposition asserts that if _a_ = _a'_, _b_ = _b'_, and _c_ = _c'_, then [L]_A_ = [L]_A'_ and the triangles are congruent.