Question

use the diagram.
write an inequality to describe the possible values of x
\\( \square < x < \square \\)

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 is larger, then the third side is longer. Here, in \(\triangle ABC\) and \(\triangle DEF\), \(AC = DE\), \(BC = EF\). The angle in \(\triangle ABC\) is \(50^{\circ}\), and the angle in \(\triangle DEF\) is \(180-(5x - 10)\) degrees.

Step2: Set up the inequalities

Since \(AB = 12\) and \(DG=10\), by the Hinge Theorem:
\(0<180-(5x - 10)-50<180\) (the difference of angles must be positive and less than \(180^{\circ}\)).
First, simplify \(180-(5x - 10)-50\):
\(180-(5x - 10)-50=180 - 5x+10 - 50=140 - 5x\)
So, \(0<140 - 5x\) and \(140 - 5x<180\)
For \(0<140 - 5x\), we get \(5x<140\), \(x < 28\)
For \(140 - 5x<180\), we get \(- 5x<180 - 140\), \(-5x<40\), \(x>- 8\). But also, since \(180-(5x - 10)>0\) (angle measure must be positive), \(190-5x>0\), \(5x<190\), \(x < 38\). And considering the non - degenerate triangle condition for the angle related to \(x\): \(180-(5x - 10)>0\) and \(5x-10>0\) (angle measure must be positive). From \(5x-10>0\), we have \(x > 2\)

Combining with the Hinge - Theorem based inequality from the side lengths:
Since \(AB = 12\) and \(DG = 10\), using the Hinge Theorem formula \(|AB - DG|<|AC|\times| \text{angle difference}|\) (in the context of triangle side - angle relationships). The correct application of the Hinge Theorem for two triangles \(\triangle ABC\) and \(\triangle DEF\) (where \(AC = DE\), \(BC=EF\)) gives \(|12 - 10|<|BC|\times|50-(180-(5x - 10))|\) (simplifying, we use the fact that for two sides \(a,b\) of two triangles with two sides equal \(a_1=a_2\), \(b_1 = b_2\), if \(c_1>c_2\) then the included angle \(\theta_1>\theta_2\)).
The proper Hinge - Theorem inequality: \(12>10\) implies \(50>180-(5x - 10)\) (since \(AB = 12\), \(DG = 10\), \(AC = DE\), \(BC = EF\)).
\(50>180-(5x - 10)\)
\(5x-10>180 - 50\)
\(5x-10>130\)
\(5x>140\)
\(x > 28\)

Also, for the angle \(180-(5x - 10)\) to be positive: \(180-(5x - 10)>0\), \(190-5x>0\), \(x<38\)

Answer:

\(28