I think that you need to  state very clearly what you mean by  triangulation of the manifold, and what do you mean  by   "angles".  (On a manifold that would require a choice of metric.) If by triangulation you mean the affine realizeation ofba finite  simplicial complex     inside some Euclidean space, and you measure the angles using the induced metric then the answer is no. 

Consider  the  case $n=4$. The  affine  simplicial complex with $4$ vertices and homeomorphic to a manifold    is given by the boundary of a $3$-dimensional  simplex.    In this case the curvature can only be positive, and clearly one can produce metrics on the $2$-sphere that are negative somewhere.

On the other hand, if you allow the faces of this tetrahedron to be curved  then the answer could be positive. Here is one plausible solution.  

On $S^2$ fix  four points $p_1,\dotsc, p_4$ and a metric  whose curvature  at $p_i$ is $f(p_i)$. Such a metric exists  by Kazhdan-Warner.  Next connect   the points by geodesic arcs, and assume you can do this  so that these arcs  do not intersect in the interior. You now have a curved  triangulation with the properties you desire.

I think that the question needs to be formulated more carefully.