Have it tested. However, in the interim, there are several manufacturers of ceiling tiles that never used asbestos, and state so in their corporate disclosures. Armstrong is one of those that comes to mind. Typically, on the back of the tile, you have the manufacturers name, and part number, although on tiles this old, the ink may be impossible to read.
You may find the link in this answer of help in doing some additional research.
For asbestos testing, you can try these local companies:
Or you can use a national firm and mail it. Testing will range from $50 to $75, local or national.
Stay safe and do not disturb the material in anyway (don't create dust from the tile) until you've had it tested.
Hope this is helpful.
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