Geometry

Question Analysis

The main focus is to use the Hinge Theorem to find the range of possible values of . Given the angles and side - length expressions in  and , we apply the theorem to set up inequalities for .

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, then the side opposite the larger included angle is longer.

Step - by - Step Solution

1. In  and ,  (marked as congruent) and  is common.

The included angle ∠URT = 42° and .

Since ∠STR >∠URT, by the Hinge Theorem, the side opposite  (which is ) is longer than the side opposite ∠URT (which is UT = 3x + 5). So, we set up the inequality  > .

Solve the inequality  > :

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

Add 7 to both sides: > 

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