The Hinge Theorem: Comparing Triangle Side Lengths Based on Angle Sizes

Hinge Theorem

The Hinge Theorem is a geometric theorem that relates the side lengths of two triangles when the included angle of one triangle is larger than the included angle of the other triangle

The Hinge Theorem is a geometric theorem that relates the side lengths of two triangles when the included angle of one triangle is larger than the included angle of the other triangle. It states that if two sides of one triangle are congruent to two sides of another triangle, and the included angle of the first triangle is larger than the included angle of the second triangle, then the third side of the first triangle is longer than the corresponding third side of the second triangle.

More formally, let’s consider two triangles, triangle ABC and triangle XYZ. If AB = XY, AC = XZ, and angle BAC > angle YXZ, then BC > YZ.

To better understand the Hinge Theorem, imagine you have two triangles that share a common side (let’s say AB = XY) and a common angle (AC = XZ). If we know that the measure of angle BAC is greater than the measure of angle YXZ, then intuitively, the lengths of BC and YZ will be different.

To see why this holds true, consider rotating triangle XYZ around point X so that sides XY and XZ overlap with sides AB and AC. The point Y will move towards B, and point Z will move towards C. Since angle BAC is greater than angle YXZ, when the two triangles overlap, point Y will end up closer to C compared to point Z ending up closer to B. Therefore, the length of BC will be greater than the length of YZ.

This theorem can be useful in various situations. For example, if you know the side lengths and measures of some angles in two triangles, and you want to determine which triangle has the longer third side, you could apply the Hinge Theorem to find the answer.

Overall, the Hinge Theorem provides a useful tool for comparing side lengths between triangles when certain conditions are met, allowing us to make conclusions about the relative lengths of their sides based on the sizes of their angles.

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