Question

determine which segment is shortest in the diagram. note that the figure to the right is not drawn to scale.
the shortest segment in the diagram is

Explanation:

Step1: Find angles in $\triangle XYZ$

In $\triangle XYZ$, $\angle Y = 38^{\circ}$, $\angle YZX=117^{\circ}$, so $\angle YXZ = 180^{\circ}-38^{\circ}-117^{\circ}=25^{\circ}$

Step2: Find angles in $\triangle XWZ$

In $\triangle XWZ$, $\angle W = 42^{\circ}$, $\angle XZW=180^{\circ}-119^{\circ}=61^{\circ}$, so $\angle WXZ = 180^{\circ}-42^{\circ}-61^{\circ}=77^{\circ}$

Step3: Apply the triangle - side relationship

In a triangle, the side opposite the smaller angle is shorter.
In $\triangle XYZ$, side opposite $\angle YXZ = 25^{\circ}$ is $\overline{YZ}$
In $\triangle XWZ$, side opposite $\angle W = 42^{\circ}$ is $\overline{XZ}$, side opposite $\angle WXZ = 77^{\circ}$ is $\overline{WZ}$, side opposite $\angle XZW = 61^{\circ}$ is $\overline{WX}$
Since $25^{\circ}<42^{\circ}<61^{\circ}<77^{\circ}<117^{\circ}$, in $\triangle XWZ$, the side opposite the smallest non - zero angle among the relevant triangles (considering the segments in question) is $\overline{WX}$ (as we are comparing the given segments). The side - angle relationship (longer side is opposite larger angle) helps us conclude that $\overline{WX}$ is the shortest.

Answer:

$\overline{WX}$