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Problem 28 Hard Difficulty

Write About It The picture shows a door hinge in two different positions. Use the picture to explain why Theorem $5-6-1$ is called the Hinge Theorem.

Answer

The two sides of the hinge represent two sides of a triangle and the distance of the opening is the third side. Since the length of the third side depends on the opening angle of the hinge, it is an appropriate name for the concept of the Hinge Theorem.

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Video Transcript

Let's take a look at the diagram that actually represents a doorway where angle B is where the hinge would be holding the door to the doorframe. You can see that the doors open at 90 degrees. At this point, if we actually attempt to close the door, the red line represents the opening of the doors. So we're closing the door and you can see that as we close the door, the red line is getting shorter, but so is the angle between the door in the doorway. So the angle measurement correlates to the measure of the triangle represented by this red dotted line. Um, as far as 1/3 side of a triangle. So what does the Hinch there, um, have to do with a doorway? So let's take a look at the hinges on the doorway. So again, angle B represents the hinge itself with two sides and segment A C represents the distance between the two sides of the hinge. And right now we've got a pair of hinges that are said at the same angle. And so, if we just looked at this as geometry triangles, you could see side a B corresponds, decide X Y side B C corresponds decide y Z, and the distance opposite of our angle are hinge is represented by Segment A C and segment X Z. So if I change the angle between the hinges and make the angle a smaller angle, so now the included angle between the two sides of the hinge is a smaller angle on our triangle on a right versus the triangle on our left, and the side opposite of our angle is a shorter measurement on the left. I can actually spread the hinge out, and the larger the angle becomes, the longer the distance between a C. Because so the hinge there of states. If you have two sides of one triangle correspond, Ercan congruent the two sides of a second triangle and the angle between the corresponding concurrent sides. The included angle is not congruent. Then we can compare their measures 86 point for the larger angle, 50 point to the smaller angle. This side opposite of the larger angle is gonna be a longer measurement than the side opposite of the smaller angle, and that is because on a hinge, the two sides that are congruent don't change in measurement on Lee, the distance between their endpoints changes

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