There are a number of theorems in elementary geometry which are true, but with (basically) one exception. Since these are some of the most basic and most cited/discussed such results and the fact that they are not universally true is rarely mentioned, it is perhaps worth posting them on this site. The most ubiquitous examples are: the pons asinorum (if two sides of a triangle are equal then so are the opposite sides), the converse of Pythagoras and the SAS theorem (if two corresponding sides and included angles of a pair of triangles are equal, then so are the remaining side and angles). Before pointing out the exceptional cases, let me start by responding to a possible accusation of nitpicking. Firstly, it is part of the mathematician's hippocratic oath that when he states a theorem, it should be true (without exception). Perhaps more importantly, a typical proof in synthetic geometry involves starting from a basic configuration, then successively constructing points, lines, triangles, circles ... , followed by making suitable deductions about the new elements. Now if, for example, one proceeds by showing that a triangle constructed in this manner has two equal sides and deducing the equality of two angles, using pons asinorum, it is then imperative to show that the exceptional case cannot occur, or, alternatively, to expand the proof to include it.
It will suffice to illustrate my point about gaps in the statements of celebrated results to consider the converse to Pythagoras: if in a triangle $ABC$, $|AC|^2=|AB|^2+|BC|^2$, then $<ABC$ is a right angle. The exceptional case is that where $A=B$, since then the above angle is not (and cannot be sensibly) defined.
Such anomalies can lead to some rather delicate problems. Such a case is the Steiner-Lehmus theorem (if two angle bisectors of a triangle are equal, then it is isosceles) which has generated much controversy, even amongst some very distinguished mathematicians, not so much for the result itself, but for the nature of the proofs, in particular, the question of whether it is possible to give a simple, direct one. Again, the case where two of the vertices of the triangle coincide is exceptional and some arguments against the existence of a direct proof use a continuity argument--varying the shape of the triangle in a way which necessarily involves passing through the dodgy situation.