Question

find the range of values for z

z + 23
3z + 9
120°

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, and the included angle of the first is larger than the included angle of the second, then the third side of the first is longer than the third side of the second.
Here, the two triangles have two congruent sides (marked with red - tick). The non - congruent included angles: one is \(120^{\circ}\) and the other is \(90^{\circ}\), and \(120^{\circ}>90^{\circ}\). So, \(3z + 9>z + 23\).

$$3z+9>z + 23$$

Step2: Solve the inequality

Subtract \(z\) from both sides:

$$3z - z+9>z - z+23$$

$$2z+9>23$$

Subtract 9 from both sides:

$$2z+9 - 9>23 - 9$$

$$2z>14$$

Divide both sides by 2:

$$z > 7$$

Also, since the length of a side of a triangle must be positive. For \(z + 23>0\) (always true for \(z>7\)) and \(3z+9>0\) (also always true for \(z>7\))

Answer:

\(z>7\)