Understanding the Hinge Theorem in Geometry Chapter 6

Hinge Theorem

The Hinge Theorem states that if two sides of one triangle are congruent to two sides of another triangle, but the included angle of the first triangle is larger than the included angle of the second, then the third side of the first triangle will be longer than the third side of the second triangle.

Think of it like opening and closing a door - the wider you open it (larger angle), the greater the distance between the edges. This theorem helps us make comparisons between triangles even when we don't have complete congruence.

Let's see this in action with an example: If $\overline{JK} \cong \overline{LK}$ (two pairs of sides are congruent) and $m\angle KJM > m\angle KLM$ (the included angle is larger in the first triangle), then $\overline{JM} > \overline{LM}$ (the third side is longer in the first triangle).

Try This! When working with the Hinge Theorem, remember to identify which angle is the "included angle" - it's the angle formed by the two congruent sides you're comparing between triangles.

The proof follows a logical sequence: we establish the congruent sides, verify the angle relationship, and then apply the Hinge Theorem to reach our conclusion about the third sides. This theorem is particularly useful when comparing distances in geometric figures.