Certain Geometry proofs seem not rigorous at all. For example, this proof from Kiselev's "Planimetry":
Theorem: The diameter (here, AB), perpendicular to a chord (here, CD), bisects the chord and each of the two arcs subtended by it.
The proof: Fold the diagram along the diameter AB so that the left part of the diagram falls onto the right one. Then the left semicircle will be identified with the right semicircle, and the perpendicular KC will merge with the perpendicular KD (where K is the point of intersection of AB and CD).
It follows that the point C, which is the intersection of the semicircle and KC, will merge with D. Therefore KC = KD, arcs BC and BD are equal, and arcs AC and AD are equal.
To me, the argument that "folding" the diagram proves that both semicircles are identified with each other is very unconvincing. It's so far from the standard of other proofs in Geometry, it seems to be almost on the level of "Well you can kind of see it, right? There you go".  
EDIT: To clarify, my question is whether or not i'm correct in thinking that this is not a rigorous proof. Am I missing something? Is the argument about folding the diagram in half actually valid? If so, why? And if not, what would be a better proof of that theorem?
 A: The way I see it there's really no objective criteria for what constitutes a "rigorous" proof. Ideally a proof would lead straight back to some of the axioms of your theory, but of course doing so for every theorem gets tedious and overly complicated really fast. It seems to me that the way proofs are used usually is more about convincing other mathematicans to believe what you're saying. From that perspective a proof is rigorous enough if it's convincing for the reader (and this varies with different audiences).$^*$
Often the problem is in translating an intuitive idea into something more formal and "rigorous" (i.e. convincing). Sometimes the author is proficient enough in his/her field to not see the need of making this translation, whereas the reader might feel that something is lacking because of this.
Anyway, in your particular case, would you feel better about replacing "fold the diagram along the diameter" with "reflect everything over the diameter"? The two things really convey the same information, but the latter might sound more "mathematical" and "proof-like".
 $^*$Of course this view has it's own problems and questions but delving into those would turn this into an essay on mathematical philosophy, and we don't want that, do we? :) 

