Question

find the smallest angle of (\triangle ghi). assume that (s) is a positive number.

Explanation:

Step1: Compare the side lengths

In a triangle, the smallest angle is opposite the shortest side. Given side lengths \(46s\), \(111s\), and the third side (let's call it \(GH\)). By triangle side - length relationships, we know that in \(\triangle GHI\), the side opposite \(\angle H\) is \(46s\), the side opposite \(\angle G\) is \(111s\), and the side opposite \(\angle I = 93^{\circ}\) is \(GH\). Since \(46s<111s\) and using the law of sines \(\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}\) (where \(a,b,c\) are side lengths and \(A,B,C\) are opposite angles), the shorter side has the smaller opposite angle.

Step2: Determine the smallest - angle relationship

Let the sides of the triangle be \(a = 46s\), \(b = 111s\), and \(c=GH\). The angles opposite these sides are \(\angle H\), \(\angle G\), and \(\angle I\) respectively. Since \(a < b\), and \(\angle I=93^{\circ}\) (a non - acute angle), and in a triangle \(\angle G+\angle H+\angle I = 180^{\circ}\), \(\angle I = 93^{\circ}\), so \(\angle G+\angle H=180^{\circ}- 93^{\circ}=87^{\circ}\). Also, from the side - angle relationship (longer side has a larger opposite angle), the angle opposite the shorter side \(46s\) (which is \(\angle H\)) is the smallest.

Answer:

\(\angle H\)