Question

determine which segment is shortest in the diagram.
choose the shortest segment below.
a. \\( \overline { w x } \\)
b. \\( \overline { w y } \\)
c. \\( \overline { x y } \\)
d. \\( \overline { w z } \\)
e. \\( \overline { y z } \\)

Explanation:

Step1: Find the angles in \(\triangle WYZ\)

In \(\triangle WYZ\), we know one angle is \(95^{\circ}\) and another is \(38^{\circ}\). Using the angle - sum property of a triangle (\(\angle W+\angle Y+\angle Z = 180^{\circ}\)), we can find \(\angle WYZ\)'s adjacent angle. But for the side - angle relationship, we know that in a triangle, the side opposite the smaller angle is shorter.

Step2: Use the side - angle relationship in \(\triangle WXY\)

In \(\triangle WXY\), \(\angle X = 50^{\circ}\), \(\angle W=45^{\circ}\). The side opposite \(\angle W\) is \(XY\), the side opposite \(\angle X\) is \(WY\), and the side opposite the right - angle (if we consider the larger triangle, but focusing on \(\triangle WXY\)) is not needed here. In \(\triangle WXY\), since \(45^{\circ}<50^{\circ}\), the side opposite \(45^{\circ}\) (\(XY\)) is shorter than the side opposite \(50^{\circ}\) (\(WY\)).

Step3: Consider the overall triangle properties (side - angle relationship)

In a triangle, the length of a side is related to the measure of the angle opposite it. The smaller the angle, the shorter the side.
In \(\triangle WXY\), \(\angle W = 45^{\circ}\), \(\angle X=50^{\circ}\), and \(\angle WXY\) (the third angle of \(\triangle WXY\)) is \(180-(45 + 50)=85^{\circ}\).
The side opposite \(\angle W\) is \(XY\).

Answer:

C. \(\overline{XY}\)