Question

station 10: arrange the sides from least to greatest.
(there is a triangle with vertices labeled f, g, h. angle at g is 109°, angle at h is 30°.)

Explanation:

Step1: Find the third angle

In a triangle, the sum of interior angles is \(180^\circ\). Given \(\angle G = 109^\circ\) and \(\angle H = 30^\circ\), let \(\angle F=x\). Then \(x + 109^\circ+ 30^\circ=180^\circ\), so \(x = 180^\circ-(109^\circ + 30^\circ)=41^\circ\).

Step2: Apply the triangle side - angle relationship

In a triangle, the larger the angle, the longer the side opposite to it. The angles of \(\triangle FGH\) are: \(\angle G = 109^\circ\), \(\angle F = 41^\circ\), \(\angle H=30^\circ\). So the order of angles from least to greatest is \(\angle H<\angle F<\angle G\). The sides opposite to these angles are \(FG\) (opposite \(\angle H\)), \(GH\) (opposite \(\angle F\)), and \(FH\) (opposite \(\angle G\)) respectively. So the order of sides from least to greatest is \(FG < GH < FH\).

Answer:

The sides from least to greatest are \(FG\), \(GH\), \(FH\) (or in terms of the sides opposite the angles: side opposite \(30^\circ\) ( \(FG\) ), side opposite \(41^\circ\) ( \(GH\) ), side opposite \(109^\circ\) ( \(FH\) )).