Hinge theorem

The theorem is an immediate corollary of the law of cosines.^[2] For two triangles with sides ${\displaystyle \{a,b,c\}}$ and ${\displaystyle \{a,b,{\hat {c}}\}}$ with angles ${\displaystyle \gamma }$ and ${\displaystyle {\hat {\gamma }}}$ opposite the respective sides ${\displaystyle c}$ and ${\displaystyle {\hat {c}}}$, the law of cosines states:

$${\displaystyle {\begin{aligned}c^{2}&=a^{2}+b^{2}-2ab\cos \gamma ,\\{\hat {c}}^{2}&=a^{2}+b^{2}-2ab\cos {\hat {\gamma }}.\end{aligned}}}$$

The cosine function is monotonically decreasing for angles between ${\displaystyle 0}$ and ${\displaystyle \pi }$ radians, so ${\displaystyle {\hat {\gamma }}>\gamma }$ implies ${\displaystyle {\hat {c}}>c}$ (and the converse as well).