Hinge theorem
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In geometry, the hinge theorem (sometimes called the open mouth theorem) states that if two sides of one triangle are congruent to two sides of another triangle, and the included angle of the first is larger than the included angle of the second, then the third side of the first triangle is longer than the third side of the second triangle. This theorem is given as Proposition 24 in Book I of Euclid's Elements.
In geometry, the hinge theorem (sometimes called the open mouth theorem) states that if two sides of one triangle are congruent to two sides of another triangle, and the included angle of the first is larger than the included angle of the second, then the third side of the first triangle is longer than the third side of the second triangle. This theorem is given as Proposition 24 in Book I of Euclid's Elements. \begin{align} c^2 &= a^2 + b^2 - 2ab\cos \gamma, \\ \hat{c}^2 &= a^2 + b^2 - 2ab\cos \hat \gamma. \end{align} The cosine function is monotonically decreasing for angles between 0 and \pi radians, so \hat\gamma > \gamma implies \hat c > c (and the converse as well).
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