Triangle Inequalities 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.

Yes, because 15 + 12 > 9, 15 + 9 > 12, and 12 + 9 > 15.

The Hinge Theorem states that if two sides of one triangle are congruent to two sides of another triangle, and the included angles are not equal, then the longer side is opposite the larger angle.

BC is greater than 5.

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

Yes, because 3 + 8 > 9, 3 + 9 > 8, and 8 + 9 > 3.

AC is longer than AB.

The minimum length of the third side must be greater than 3 (10 - 7).

It is an isosceles triangle.

The maximum length of the third side must be less than 17 (5 + 12).