Suppose on a non-positive curvature Riemannian manifold,we have a geodesic triangle $\triangle abc$ ,and counterpart edges donates $\alpha,\beta,\gamma$.
If now I get
$$ a^2 \geqq b^2+c^2-2bc cos\alpha $$
How can I induce that $\alpha+\beta+\gamma \leqq \pi$?
I will appreciate your help!
Take a Euclidean triangle with sides $a,b,c$ (same as the hyperbolic triangle) and angles $\alpha',\beta',\gamma'$. Comparing your inequality with $$a^2=b^2+c^2-2bc\cos\alpha'$$yields$$\alpha\leq\alpha'.$$