Consider the following two triangles. Length 𝑎 is congruent to 𝑑. And length 𝑏 is congruent to 𝑒. Given that 𝜙 is greater than 𝜃, what does the hinge theorem tell us about the lengths 𝑐 and 𝑓?
So we’re told that various lengths in these two triangles are congruent. 𝑎 is congruent to 𝑑. And 𝑏 is congruent to 𝑒. We’re also told that 𝜙 is greater than 𝜃 and told to use the hinge theorem to answer this question. Let’s recall what the hinge theorem tells us.
It tells us that if two triangles have two equal sides, then the triangle with the larger included angle has the longer third side. The included angle is the angle between the two equal sides. So that is angle 𝜃 in the first triangle and angle 𝜙 in the second.
So the hinge theorem states that whichever triangle has the larger included angle will also have the longer third side. We already know that angle 𝜙 is greater than angle 𝜃. And therefore the third side in the second triangle will be longer than the third side in the first. So we can conclude that length 𝑓 is greater than length 𝑐.