Hinge theorem

The Hinge theorem compares two triangles with two pairs of equal corresponding sides: the triangle with the larger included angle has the longer opposite side.

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Triangle inequality: In any triangle, the length of each side is less than the sum of the other two side lengths. Its bounds help explain why changing the included angle changes the third side.
Triangle: A polygon with three sides, three vertices, and three interior angles. The theorem compares two triangles sharing equal corresponding side lengths.
Indirect proof: A proof strategy that assumes a claim is false and derives a contradiction. A common proof compares a hypothetical third side against the triangle inequality.
Converse of the Hinge theorem: A triangle comparison theorem stating that the triangle with the longer third side has the larger included angle when two corresponding sides are equal. It reverses the Hinge theorem’s conclusion to infer an angle comparison from side lengths.
Law of cosines: A formula relating a triangle’s side lengths to the cosine of one of its angles. With two fixed sides, it directly relates the included angle to the opposite side.
Corresponding parts: Parts of two figures paired by their positions or roles in a comparison. The theorem requires identifying which sides and angles correspond across the triangles.
Geometric inequality: An inequality expressing an order or bound among geometric quantities. The theorem provides an order relation between opposite sides from angle information.
SSS congruence theorem: A triangle congruence criterion stating that three pairs of corresponding sides determine congruent triangles. SSS fixes the third side, whereas the Hinge theorem compares it when only two sides match.
Included angle: The angle formed by two specified sides of a triangle or other geometric figure. This is the angle whose size determines which opposite side is longer.
Angle: A figure formed by two rays or line segments sharing an endpoint, measured by their separation. The included angle is the comparison that determines the opposite-side ordering.