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:

Step1: Apply the Hinge Theorem

The Hinge Theorem states that if two sides of one triangle are congruent to two sides of another triangle, but the included angle of the first triangle is larger than the included angle of the second triangle, then the side opposite the larger included angle is longer.
Here, \(TR = GE\), \(SR = FE\), and \(\angle R=56^{\circ}\), \(\angle E = 42^{\circ}\). Since \(56^{\circ}>42^{\circ}\), we have \(TS>GF\).

Step2: Compare the values

Given \(GF = 3.2\) ft.

• For \(1.6\) ft: \(1.6<3.2\), so it is not valid.
• For \(3.0\) ft: \(3.0<3.2\), so it is not valid.
• For \(3.2\) ft: \(3.2 = 3.2\), so it is not valid.
• For \(4.0\) ft: \(4.0>3.2\), so it is valid.

Answer:

\(4.0\) ft