What computer languages do Linux-based OSes use?
Linux (the kernel) is essentially written in C with a little of assembly code.
The lower layer of userland, usually GNU (glibc and other libraries plus standard core commands) are almost exclusively written in C and shell scripting.
The remaining of the Gnu/Linux distributions userland is written in any language developers decide to use (still a lot of C and shell but also C++, python, perl, javascript, java, C#, golang, whatever ...)
I believe the following link has the answer you're looking for: http://ubuntuforums.org/showthread.php?t=705754
To paraphrase my favorite answers there:
Linux, the kernel, is mostrly written in C and a little bit of Assembly.
Toolkits and frameworks used to develop the graphical interface (e.g. GTK+, Qt, GNOME, KDE, Unity) are written mostly in C and C++.
Utilities and applications (i.e. programs) that come bundled with the Operating System are usually written in Java, Python, C, C++ and even C#.