Triangle Inequalities and Hinge Theorem Review

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 length of the third side must be greater than 1 (3 - 4) and less than 7 (3 + 4). So, the range is (1, 7).

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.

$$m\angle1 > m\angle2$$ .

The longest side is 13.

$$AB < DE$$ .

Yes, it satisfies the theorem because 7 + 10 > 12, 7 + 12 > 10, and 10 + 12 > 7.

No, it cannot because 2 + 2 is not greater than 5.

The larger the angle, the longer the opposite side.

$$m\angle1 > m\angle2$$ .