Question

the angle measures of △abc are a = 60°, b = 59°, and c = 61°. list the sides of the triangle in order from shortest to longest.

Explanation:

Step1: Recall the triangle side-angle relationship

In a triangle, the larger the angle, the longer the side opposite to it. This is based on the Law of Sines or the basic triangle side - angle relationship (in a triangle, if \(\angle X>\angle Y\), then the side opposite \(\angle X\) (\(x\)) is longer than the side opposite \(\angle Y\) (\(y\)).

Step2: Order the angles from smallest to largest

Given the angles of \(\triangle ABC\): \(\angle B = 59^{\circ}\), \(\angle A=60^{\circ}\), \(\angle C = 61^{\circ}\). So, the order of angles from smallest to largest is \(\angle B<\angle A<\angle C\).

Step3: Determine the order of the sides

The side opposite \(\angle B\) is \(AC\), the side opposite \(\angle A\) is \(BC\), and the side opposite \(\angle C\) is \(AB\). Since the side opposite a smaller angle is shorter, the order of the sides from shortest to longest (based on the order of their opposite angles) is: the side opposite \(\angle B\) ( \(AC\) ), the side opposite \(\angle A\) ( \(BC\) ), the side opposite \(\angle C\) ( \(AB\) ). Or in terms of the standard notation where side \(a\) is opposite \(\angle A\), side \(b\) is opposite \(\angle B\), side \(c\) is opposite \(\angle C\). So \(b\) (opposite \(\angle B\)), \(a\) (opposite \(\angle A\)), \(c\) (opposite \(\angle C\)). Since \(\angle B = 59^{\circ}\), \(\angle A = 60^{\circ}\), \(\angle C=61^{\circ}\), the order of sides (from shortest to longest) is \(b\) ( \(AC\) ), \(a\) ( \(BC\) ), \(c\) ( \(AB\) ) or using the angle - side correspondence: side opposite \(59^{\circ}\) ( \(AC\) ), side opposite \(60^{\circ}\) ( \(BC\) ), side opposite \(61^{\circ}\) ( \(AB\) ).

Answer:

\(AC\), \(BC\), \(AB\) (or in standard triangle notation, \(b\), \(a\), \(c\) where \(a = BC\), \(b = AC\), \(c = AB\))