### Video Transcript

Consider triangles π΄π΅πΆ and π·πΈπΉ in the figure. Without completing any calculations, use the Hinge theorem to determine whether π·πΉ is greater than, less than, or equal to π΄πΆ.

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. Letβs consider this one step at a time. The two triangles have two equal length sides of size three and four. The length π΅πΆ is equal to the length πΈπΉ, and the length π΄π΅ is equal to the length π·πΈ. This means that two sides of the triangles are congruent.

The included angle of the first triangle, angle π΅, is equal to 42 degrees. The included angle of the second triangle, angle πΈ, 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 π·πΉ will be greater than the length of π΄πΆ. Using the Hinge theorem, we can conclude in this case that the length π·πΉ is greater than the length π΄πΆ.