The hinge theorem

After the equalities (the congruences), we study the inequalities between the elements of a triangle. The hinge theorem compares two triangles with two pairs of equal sides but a different included angle: a larger angle corresponds to a larger opposite side, exactly like a hinge that opens.

Theorem — the hinge theorem

If two triangles have two pairs of respectively congruent sides but the included angles are unequal, then the third sides are unequal in the same sense: $\begin{cases} AB\cong A'B' \\ BC\cong B'C' \\ \widehat{ABC} > \widehat{A'B'C'} \end{cases} \implies \overline{AC} > \overline{A'C'}$

The same two sides, but a larger included angle on the left: the third side $AC$ is longer than $A'C'$.

The converse theorem also holds: if the third sides are unequal, so are the included angles.

Links

Topics: Euclidean geometry
Concepts: Triangle inequality · Hinge theorem · Converse theorem · Triangle
Skills: Synthetic geometry