Is it okay to have no heater for my 10 gallon planted tank? Will it cause harm to them?
What kind of plants do you have in there? Any fish/shrimp?
If you have fish that don't need a heater or no fish/inverts, it'll be fine. Plants can tolerate a much wider range of temperatures than fish and other livestock (with a few exceptions of course)
As long as your room temperature stays within the comfortable range for the plants, then it should be fine.
I have currently have no fish in there but will be getting some after i finish the cycle once i receive my heater and filter in the mail probably 2 weeks from now.
My average temperature inside the house is 75-82 degrees. Is that okay?
That's comfortable for most plants, yes
Does your temperature always stay the same or do you have seasons where it will be really cold, really hot, and so forth? You need to check the temp. of your water, not just the house. I personally always have a heater with tropical fish.
OP has a heater coming in the mail, and no fish until the heater gets here and cycle finishes. They're only asking about plants, at the moment.
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