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 \(\triangle YXZ\)

In \(\triangle YXZ\), we know \(\angle YZX = 180^{\circ}-(38^{\circ}+117^{\circ})=25^{\circ}\) (using the angle - sum property of a triangle \(\angle A+\angle B+\angle C = 180^{\circ}\)).

Step2: Find angles in \(\triangle XWZ\)

In \(\triangle XWZ\), \(\angle XZW=180^{\circ}-(119^{\circ}+42^{\circ}) = 19^{\circ}\) (using the angle - sum property of a triangle \(\angle A+\angle B+\angle C = 180^{\circ}\)).

Step3: Recall the side - angle relationship

In a triangle, the shorter side is opposite the smaller angle.

• In \(\triangle YXZ\), sides are \(XY\), \(YZ\), \(XZ\) with opposite angles \(\angle YZX = 25^{\circ}\), \(\angle YXZ=38^{\circ}\), \(\angle XYZ = 117^{\circ}\) respectively. So \(XZ
• In \(\triangle XWZ\), sides are \(XW\), \(WZ\), \(XZ\) with opposite angles \(\angle XZW = 19^{\circ}\), \(\angle WXZ=119^{\circ}\), \(\angle XWZ = 42^{\circ}\) respectively. So \(XZ

Answer:

\(\overline{XZ}\)