Quick answer - no. Each GL_TRIANGLES element is independent from the previous triangle. However, trying to reuse the same veritces in the following triangle sometimes allows the low level driver/hardware to speed up the triangle rendering by converting the GL_TRIANGLES into a GL_TRIANGLE_STRIP - but is heavily dependent on the hardware and order. Google for it if you want more info.
Do a little research on what each one does. GL_TRIANGLES will create a triangle from each 3 points you give it, so you can wrap them all between a glBegin and glEnd.
See this page for more info on them and other primitives.