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htm should be Hypertext Markup (basically html, so any web browser should open). From google, *.mth appears to be Derive Math File, basically arithmetic operations allowed by a program. Not sure if that helps you at all. Try opening it with a text editor first.
What subject is this anyway?
EDIT: more google seems to indicate that Derive was an algebra program for Windows that was discontinued in 2007.
Pardon me if this sounds dumb, but are you sure it's not just another *.htm file and the instructor just typoed on the file extension? Have you already tried opening it in a text editor as has been suggested?
You should be aware that Linux (like almost all UNIX based systems) does not use a "file extension" convention to identify file types. Each program reads its input files and looks for a "magic number" in the file which tells it what type of input is in the file. If there is no "magic number," the file is assumed to be a text file.
If a program expects a specific type of input file, it will (usually) report an error if it finds the wrong input file type.
The file command, noted above, reports the meaning of a file's "magic number".
Since you are working on an "Art Appreciation" class, I'd try opening the file with gimp in the hope that it was some type of picture format that gimp could read.
You should be aware that Linux (like almost all UNIX based systems) does not use a "file extension" convention to identify file types. Each program reads its input files and looks for a "magic number" in the file which tells it what type of input is in the file.
One thing I've noticed is that apparently not all programs use the "magic number" method...graphical file managers tend to use extensions if they're provided.
I create an empty file called "something.png" using touch.
I give the file contents, e.g. the text "The quick brown fox jumps over the lazy dog."
I view the file listing in a graphical file manager (e.g. Thunar in Xfce), and it lists its type as a PNG image, when it's actually a plain text file.