Let $(M,g)$ be a complete Riemannian manifold with sectional curvature $\sec_M\ge k$. Let $\gamma_1:[0,a]\to M$ and $\gamma_2:[0,b]\to M$ be minimizing geodesic segments issuing from $p$ and forming angle $\alpha$. Let $\bar{\gamma}_1$ and $\bar{\gamma}_2$ be the hinge in $M_k^2$ with the same adjacent side lengths $a,b$ and included angle $\alpha$. When $k>0$, assume

\begin{align*} a<\frac{\pi}{\sqrt{k}}, \qquad b<\frac{\pi}{\sqrt{k}}, \end{align*}

and assume that the model hinge is chosen in the comparison range where the opposite side

\begin{align*} \bar{c}=d_{M_k^2}(\bar{\gamma}_1(a),\bar{\gamma}_2(b)) \end{align*}

exists uniquely, satisfies $\bar{c}<\pi/\sqrt{k}$, and gives a model triangle of perimeter $a+b+\bar{c}<2\pi/\sqrt{k}$. Also assume, in the positive-curvature case, that the actual opposite side

\begin{align*} c=d_M(\gamma_1(a),\gamma_2(b)) \end{align*}

satisfies $c<\pi/\sqrt{k}$ and $a+b+c<2\pi/\sqrt{k}$, so that the actual geodesic triangle lies in the spherical Toponogov comparison range. Then

\begin{align*} d_M(\gamma_1(a),\gamma_2(b)) \le d_{M_k^2}(\bar{\gamma}_1(a),\bar{\gamma}_2(b)). \end{align*}