Triangle Inequality and Hinge Theorem

The Triangle Inequality Theorem states that for any triangle, the sum of the lengths of any two sides must be greater than the length of the third side.

The Hinge Theorem 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 third side of the second triangle.

No, because 3 + 4 is not greater than 8.

$$2 < x < 8$$ .

$$5 < x < 15$$ .

5 is the shortest side.

The smallest angle is opposite the shortest side, which is 6.

In a triangle, the larger the side, the larger the opposite angle.

The third side must be greater than 3 and less than 17, so $$3 < x < 17$$ .

The Hinge Theorem helps determine which triangle is larger based on the lengths of two sides and the included angle.