Geometry

Question Analysis

The main focus is applying the Hinge Theorem to find the range of possible values of . Given two triangles  and  with congruent - side pairs 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 , UR = ST (marked as congruent) and  is common.

The side UT = 10 and . Since  > , by the Hinge Theorem, the angle opposite  (which is ) is larger than the angle opposite UT (which is ). So, we set up the inequality  < .

Solve the inequality  < :

Subtract 9 from both sides:

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