Yes and no.
As others have already answered, the language C++ is a superset of the language C, with some small exceptions, for example that sizeof('x') gives a different value.
But what I don't think has been very clearly stated is that when it comes to the use of these two languages, C++ is not a superset, but rather different. C++ contains new (it can be discussed if they are better) ways of doing the basic things, such as writing to the screen. The old C ways are still there, but you generally use the new ways. This means that a simple "hello world" program looks different in C and in C++. So it is not really true that the simple things are the same in C and C++, and then you just add more advanced stuff, such as support for object-oriented programming, in C++.
So if you have learnt C++, you will need to re-learn quite a lot before you can program in C. (Well, it is possible to teach C++ as an extension to C, still using printf and malloc instead of iostreams and new, and then adding classes and other C++ things, but that way of using C++ is generally frowned upon.)