Which statement reflects the Hinge Theorem?

Which statement reflects the Hinge Theorem?

• A. If two triangles have two corresponding sides congruent and the included angle of the first is larger, then the third side of the first is longer
• B. If two triangles have two corresponding sides congruent and the included angle of the first is smaller, then the third side of the first is longer
• C. If two triangles have two corresponding sides congruent and the included angle is equal, the third sides are equal
• D. If two triangles have two corresponding sides not congruent, nothing can be said about the third sides

Answer: (A)

The hinge theorem says that if two triangles share two congruent sides, the one with a larger angle between those sides has the longer third side. Picture two fixed-length segments joined at a common vertex; widening the angle between them pushes their outer endpoints farther apart, so the distance between those endpoints—the third side—gets bigger. Algebraically, if the two fixed sides are a and b in both triangles and their included angles are C and C', then c^2 = a^2 + b^2 - 2ab cos C and c'^2 = a^2 + b^2 - 2ab cos C'. When C > C', cos C < cos C', making c > c'. This matches the statement that the first triangle’s third side is longer when its included angle is larger, given the two corresponding sides are congruent. If the included angle were equal, SAS would make the triangles congruent and their third sides equal, which is a related special case but not the comparison the hinge theorem specifically describes.

The hinge theorem says that if two triangles share two congruent sides, the one with a larger angle between those sides has the longer third side. Picture two fixed-length segments joined at a common vertex; widening the angle between them pushes their outer endpoints farther apart, so the distance between those endpoints—the third side—gets bigger. Algebraically, if the two fixed sides are a and b in both triangles and their included angles are C and C', then c^2 = a^2 + b^2 - 2ab cos C and c'^2 = a^2 + b^2 - 2ab cos C'. When C > C', cos C < cos C', making c > c'.

This matches the statement that the first triangle’s third side is longer when its included angle is larger, given the two corresponding sides are congruent. If the included angle were equal, SAS would make the triangles congruent and their third sides equal, which is a related special case but not the comparison the hinge theorem specifically describes.