# Comparing Triangles with the Hinge Theorem

Instructor: Laura Pennington

Laura has taught collegiate mathematics and holds a master's degree in pure mathematics.

The hinge theorem is a useful theorem in real-life applications. This lesson will define the hinge theorem and use it to compare triangles. We will also look at a real-life application of the hinge theorem.

## The Hinge Theorem

Suppose you and your friend, Mary, are walking through a haunted house at an amusement park, and you come across a trap door on the ground that you have to go through. You go first by opening the door so that the length of the opening is large enough for you to fit through it. Mary goes second, and has to open the door a bit wider to make the length of the opening large enough for her to fit through.

Do you notice that when the opening length is shorter, the angle at the hinge of the door is smaller than when the opening length is larger? Do you also notice that the door length and the floor length of the door remain the same in both cases, and it's just the opening length and the hinge angle that change? This has great mathematical significance! So great that there is a theorem that explains this phenomenon, and it is appropriately called the hinge theorem.

The hinge theorem states that if two triangles have two congruent sides (sides of equal length), then the triangle with the larger angle between those sides will have a longer third side. This also gives way to the converse of the hinge theorem, which states that if two triangles have two congruent sides, then the triangle with the longer third side will have a larger angle opposite that third side.

To illustrate this, think about the trap door again. When we open it, we create a triangle. One side is the door, one side is the floor length of the door, and the third side is the opening length. The wider you open the door, the greater the hinge angle and the greater the opening length.

When it's put like that, it seems like common sense! Let's take a look at using this theorem to compare triangles and in an application!

## Comparing Triangles Using the Hinge Theorem

Suppose we have two triangles as shown in the image.

It is indicated that side AB is congruent to side DE and side BC is congruent to side EF. We are also given that ∠ABC has measure 63 degrees and ∠DEF has measure 82 degrees. Based on this, what can we conclude about how sides AC and DF compare? If you are thinking that the hinge theorem tells us that DF is longer than AC, then you are correct! Because the two triangles have two congruent sides, and the angle between those sides is larger in ΔDEF, the hinge theorem tells us that ΔDEF will have a longer third side, so DF is longer than AC.

Now, consider another two triangles shown in the image.

This time, we are given that the two triangles have two congruent sides, and that AC has length 7 inches and DF has length 6 inches. We can use the converse of the hinge theorem to conclude that ∠ABC is larger than ∠DEF, because ∠ABC is opposite the longer third side.

Pretty neat, huh? Let's make it even more interesting by looking at an application!

## Application of the Hinge Theorem

Suppose there are two airplanes en route to the same airport as shown in the image.

The image shows that Airplane 1 is coming in from the east, and is 181 miles from Simpson City, and Airplane 2 is coming in from the west and is 181 miles from Simpson City in a different direction. Simpson City is 100 miles directly north from the airport. Therefore, we can create two triangles (ΔABC and ΔABD) where side AB is in both triangles, so obviously congruent to itself, and BC and BD are congruent because they have an equal length of 181 miles. We are also given the included angles of these congruent sides as ∠ABC = 48 degrees and ∠ABD = 113 degrees.

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