Triangles are made of three sides connected by angles. The angles act as "hinges" to hold two sides of the triangle together. The Hinge Theorem for triangles provides a useful way to look at the relationship between the sides and angles of a triangle.
Hinge Theorem: Given \(\mathsf{ \Delta{}}\)RSL and \(\mathsf{ \Delta{}}\)WYX, with RS ≅ WY, SL ≅ YX, and included ∠S has greater measure than included ∠Y, then RL > WX.
The converse of the theorem states the opposite: If two triangles ABC and DEF have two pairs of sides that are congruent with BC > EF, then the included angle of \(\mathsf{ \Delta{}}\)ABC is greater than the included angle of \(\mathsf{ \Delta{}}\)DEF. (m∠A > m∠D)
If a real-life situation resembles what is described in the Hinge Theorem, or its converse, then you can use the theorem to solve the problem. The Hinge Theorem is especially useful for comparing distances traveled by two objects moving on similar paths. Can you see how could the Hinge Theorem could be used to answer the following question?
Two separate walking tour groups leave from the same location and walk in opposite directions. Group A walks 4.5 miles east, then turns 45 degrees north and walks for 3 miles. Group B walks 4.5 miles west, then turns 20 degrees to the south and continues for 3 miles. Even though each group has walked 7.5 miles, one group has moved farther away from the starting point and may be out of cell phone range. Which group is farther away?
Watch the video below to see how you might approach the problem, using the Hinge Theorem to find a solution.
As you watch this video, use the study guide to follow along if you'd like. Click the button below to download the study guide.
In this video, we will walk through another real-life use of the Hinge Theorem. In this case, we have two tour groups walking from the same starting location. Group A walks 4.5 miles east, turns 45 degrees towards the north, and then walks for another 3 miles. Group B walks 4.5 miles west, turns 20 degrees south and walks for another 3 miles. Notice both groups walk a total of 7.5 miles, but one group is farther from the starting location than the other. Which is it?
In order to solve this, I suggest starting with a sketch of a diagram. To begin, let's remind ourselves of the directions for North, South, East and West… Then we can identify the starting location… Group A traveled 4.5 miles east, that's towards the right of our page… Then they turn 45 degrees north, so I'll use a protractor to get that measurement… and then we can draw another arrow in this direction to show this movement... The end of this arrow represents the location of Group A at the end of their walk. Now, pause the video, and recreate the route that Group B travelled. Be sure to use the protractor when needed. Resume playback in a moment to check your work...
When we look at the completed diagram, it should be easy to see that Group B is farther from the starting location. The members of each group don't have the benefit of seeing the route from overhead like we do, and they have no way of measuring the distance back to the starting point, so they could use the Hinge Theorem to decide which group is further from the starting point. If you remember, the Hinge Theorem states: if two sides of one triangle are congruent to two sides of another triangle, and the included angle of the first is larger than the included angle of the second, then the third side of the first triangle is longer than the third side of the second.
So, by creating triangles out of the path each group took, we see the interior angle for group A is 135 degrees, and the interior angle for group B is 160 degrees… Therefore the length of the third side of the triangle formed with Group B's path is longer than Group A. Group B is farther from the starting point.
Question
For a Hinge Theorem problem, how do you tell the difference between using the main theorem or the converse of the theorem?
