Question

which lists the sides of △xyz from shortest to longest?
(image of triangle xyz with angle at x: 58°, angle at z: 63°)
a. $overline{yz}$, $overline{xz}$, $overline{xy}$
b. $overline{xy}$, $overline{xz}$, $overline{yz}$
c. $overline{xz}$, $overline{yz}$, $overline{xy}$
d. $overline{xy}$, $overline{yz}$, $overline{xz}$

Explanation:

Step1: Find the third angle

In a triangle, the sum of interior angles is \(180^\circ\). Let the measure of \(\angle Y\) be \(y\). Then \(y + 58^\circ+ 63^\circ= 180^\circ\). So \(y = 180^\circ-(58^\circ + 63^\circ)=180^\circ - 121^\circ = 59^\circ\).

Step2: Relate angles to sides

In a triangle, the larger the angle, the longer the side opposite to it.

• Opposite \(\angle X = 58^\circ\) is \(YZ\).
• Opposite \(\angle Z = 63^\circ\) is \(XY\).
• Opposite \(\angle Y = 59^\circ\) is \(XZ\).

Now, order the angles from smallest to largest: \(58^\circ< 59^\circ< 63^\circ\) (i.e., \(\angle X<\angle Y<\angle Z\)). So the sides opposite (from shortest to longest) will be \(YZ < XZ < XY\) (since side opposite smaller angle is shorter).

Answer:

A. \(\overline{YZ}\), \(\overline{XZ}\), \(\overline{XY}\)