Darn, I was hoping for some more clear cut "change" in conditions. That would have been easier to figure out. I don't think the temperature drop you mentioned would have affected it. You can go ahead and cut off the sick looking parts if you want. Just know that this plant will "bleed" white latex. It is mildly toxic, but as long as you don't get it in your eyes or mouth don't worry about it. As long as you wash your hands afterward it shouldn't affect your skin.
That is probably only going to be a cosmetic fix. Check and see how stable the plant is in the pot. Basically just rock the stem back and forth a bit and see if it seems well rooted. If it seems pretty stable (particularly if the pot is moving with it) that is a good sign. If it seems really loose, almost like you could lift it right out of the pot like a stick that is bad as it means the roots have rotted.
Check the stems closely (especially near the tips) and look for a very fine and minute spider webbing (you won't see it unless you look very close). If you see some it is likely infested by mites. I don't really think that is the problem, but it is possible.
I've been holding off on the thing you probably won't be happy about doing, but you may need to take it out of its pot and inspect the roots. Try the other stuff first, but the thing above about rocking the plant was a quick test to save you this step, but only if things are so bad the roots have rotted out completely. Otherwise you have to do this. You are looking for healthy roots. I'm assuming you may not be a plant person, so this can be hard to describe, but in short look for nice white or pale yellow roots (mostly near the tips). If all you see are brown and somewhat mushy roots they are dead. While you are at it look for what appears to be a white fluffy fungus on the roots. If you see it you may have root mealybugs. It is an insect, but it looks fungus like unless you look closely (Google it). If you have those let us know because it means you'll need to use pesticides and we can help you with that.
Hopefully other people will have suggestions in case I'm not thinking of something.