Geometry

Question Analysis

The main focus is to use the Hinge Theorem to find the range of possible values of . Given two triangles  and  with congruent sides and different included angles, we'll set up and solve inequalities based on the theorem.

Key Concept Explanation

The Hinge Theorem states that if two sides of one triangle are congruent to two sides of another triangle, but the included angles are different, the side opposite the larger included angle is longer.

Step - by - Step Solution

1. In  and ,  and  is common.

The included angle  and .

Since  > , by the Hinge Theorem, the side opposite  (which is ) is longer than the side opposite  (which is ). So, we set up the inequality  > .

Solve the inequality  > :

Subtract  from both sides:  > , which simplifies to  > .

Add 9 to both sides:

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