Let $M$ be a (second-countable topological) compact connected manifold-with-boundary. Suppose $\partial M$ has a triangulation. Does there exist a triangulation of $M$ which extends the triangulation on $\partial M$? If not, what additional conditions would be required for this to hold?
For background, I would like to extend the result that this is true when $M$ is the closed ball in $\mathbb{R}^n$. Here the triangulation is extended by adding simplex-cones from the center of the ball to the boundary-simplices.