libbgi.a is a gcc library. It will not work with Visual Studio.
If you have to use winbgi as a course requirement, switch your compiler tools to mingw 5.0.3. Note that this uses a version of GCC that is is about 4 major versions and more than 10 years behind what is currently available.
Winbgi is an attempt to re-implement a graphics library that was popular in the late 1980s and early 1990s. It's been out of general use for over 20 years, so it isn't applicable to anything you are likely to encounter in a professional programming environment.
Note that this means your instructor is preparing you to work in a fast-paced, cutting-edge work environment by teaching you with technologies that have been obsolete for between 10 and 25 years. I strongly recommend supplementing your education with some of the materials from the C++ Book List.
If you are using this by your own choice, I recommend using a a more modern graphics library like those included in SFML (https://www.sfml-dev.org/) and SDL (https://www.libsdl.org/). They are better supported on modern systems and more closely related to the tools currently used in industry.
Edit
Digging a bit deeper, there are ports to Visual Studio 2005, 2008, and 2010.