Let $M$ be a either the Euclidian or the hyperbolic 2-plane. Given two triangles $ T = (x,y,z) $ and $ T' = (x',y',z') $, the natural map is the map which maps $ x \mapsto x' $, $ y \mapsto y' $, $ z \mapsto z' $ and who maps affinely each geodesic side of $T$ into the corresponding side of $T'$. We say that the triangle $ T $ dominate the triangle $ T' $ if the natural map is a short map (Lipschitz with constant $1$).

My question is, given a triangle $T$ with sides of length $(a, b,c)$, is it true that the triangle $T'$ with sides of length $(a+\epsilon, b+\epsilon, c+\epsilon)$ dominates $T$ for all small enough $\epsilon>0$ ?

I can prove this statement when $M$ is the Euclidian plane with some calculations involving the law of cosines, but I couldn't manage to do the same with the hyperbolic plane.

Given two (Euclidean or hyperbolic) triangles $ T = ABC $ and $ T' = A'B'C' $, the natural map is the one that sends $ A' \mapsto A $, $ B' \mapsto B $, $ C' \mapsto C $ and maps affinely each side of $T'$ onto the corresponding side of $T$. We say that the triangle $ T' $ dominates the triangle $ T $ if the natural map is a short map (Lipschitz with constant $1$) with respect to the distance in the (Euclidean or hyperbolic) plane.

My question is, given a triangle $T$ with side lengths $(a, b,c)$, is it true that the triangle $T'$ with side lengths $(a+\epsilon, b+\epsilon, c+\epsilon)$ dominates $T$ for all $\epsilon>0$ small enough?

I can prove this statement for Euclidean triangles by some calculations involving the law of cosines, but I couldn't manage to do the same in the hyperbolic plane.