Rule

Converse Hinge Theorem

If two sides of a triangle are congruent to two sides of another triangle, the triangle with the larger third side also has the larger included angle.

Based on the diagram above, the following relation holds true.

This theorem is the converse of the Hinge Theorem.

Proof

Consider and such that and where

This theorem can be proven by contradiction. Since the goal is to prove that the opposite statement will be assumed, that is, Because this is a non-strict inequality, both and have to be considered.

If then, and share two congruent sides and their included angles are congruent. Because of the Side-Angle-Side Congruence Theorem, they are congruent.
Since the triangles are congruent, their sides are congruent as well. This means that but this contradicts the fact that the given triangle is such that

If then the Hinge Theorem states that but this also contradicts the fact that

Conclusion

The assumption that is less than or equal to contradicts the hypothesis. Therefore, this assumption must be false. Consequently, the initial conclusion of the theorem is true.

Exercises