Question

in the triangles, tr = ge and sr = fe. if (overline{gf}=3.2) ft, which is a possible measure of (overline{ts})? 1.6 ft 3.0 ft 3.2 ft 4.0 ft

Explanation:

1. Recall the Hinge - Theorem:

• 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.
• In \(\triangle STR\) and \(\triangle GFE\), we know that \(TR = GE\) and \(SR=FE\), and \(\angle R = 56^{\circ}\), \(\angle E=42^{\circ}\). Since \(\angle R>\angle E\), by the Hinge - Theorem, \(TS > GF\).

1. Analyze the given options:

• Given that \(GF = 3.2\) ft. We need to find a value for \(TS\) such that \(TS>3.2\) ft.
• Among the options:
• Option A: \(1.6\) ft is less than \(3.2\) ft.
• Option B: \(3.0\) ft is less than \(3.2\) ft.
• Option C: \(3.2\) ft is equal to \(3.2\) ft.
• Option D: \(4.0\) ft is greater than \(3.2\) ft.

Answer:

D. 4.0 ft