If f(n)=θ(g(n)) and g(n)=O(h(n)), then which of the following is true?

From $f = \Theta(g)$ there is a constant c with $f(n) \le c\,g(n)$ for large n, and from $g = O(h)$ a constant d with $g(n) \le d\,h(n)$. Chaining them gives $f(n) \le cd\,h(n)$, so $f(n) = O(h(n))$. A tight bound does not follow, because h may grow much faster: with $f = g = n$ and $h = n^2$, f is $O(h)$ but not $\Theta(h)$, and h is not $O(f)$.