Question

if possible, write an inequality to describe the restrictions on x and which theorem was used to draw this conclusion. none of these answers 0 < x < 4 by the hinge theorem converse -2 < x < 4 by the hinge theorem 0 < x < 4 by the hinge theorem

Explanation:

Step1: Recall hinge - theorem concept

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.

Step2: Identify congruent sides

In the two triangles, we have two pairs of congruent sides (the marked equal - length sides).

Step3: Apply the hinge theorem

The angle in the first triangle is \(52^{\circ}\) and in the second is \(57^{\circ}\). By the hinge theorem, since \(52^{\circ}<57^{\circ}\), we have \(2x + 4<12\). Also, since side - length must be non - negative, \(2x+4>0\).

Step4: Solve the inequalities

For \(2x + 4<12\), subtract 4 from both sides: \(2x<8\), then \(x < 4\). For \(2x+4>0\), subtract 4 from both sides: \(2x>-4\), then \(x>0\). So \(0 < x < 4\).

Answer:

D. \(0 < x < 4\) by the hinge theorem