Consider triangles 𝐴𝐵𝐶 and 𝐷𝐸𝐹 in the figure. Without completing any calculations, use the Hinge theorem to determine whether 𝐷𝐹 is greater than, less than, or equal to 𝐴𝐶.
The Hinge theorem states that if two sides of a triangle are congruent, then the triangle with the larger included angle will have the larger third side. Let’s consider this one step at a time. The two triangles have two equal length sides of size three and four. The length 𝐵𝐶 is equal to the length 𝐸𝐹, and the length 𝐴𝐵 is equal to the length 𝐷𝐸. This means that two sides of the triangles are congruent.
The included angle of the first triangle, angle 𝐵, is equal to 42 degrees. The included angle of the second triangle, angle 𝐸, is equal to 89 degrees. The Hinge theorem tells us that the triangle with the larger included angle will have the larger third side. 89 degrees is greater than 42 degrees. This means that the length of 𝐷𝐹 will be greater than the length of 𝐴𝐶. Using the Hinge theorem, we can conclude in this case that the length 𝐷𝐹 is greater than the length 𝐴𝐶.