Question

list the angle measures of (\triangle ijk) in order from smallest to largest. assume that (y) is a positive number. (manglesquare<manglesquare<manglesquare)

Explanation:

Step1: Compare the side lengths

In a triangle, the larger the side length, the larger the angle opposite it. Given \(JK = 104y\), \(IK=35y\). Since \(y>0\), \(104y>35y\). Also, by triangle - side relationships (sum of two sides is greater than the third side and we can calculate the third side using the law of sines or the fact that in \(\triangle IJK\), we know one angle \(\angle K = 95^{\circ}\)). The side opposite \(\angle I\) is \(JK = 104y\), the side opposite \(\angle J\) is \(IK = 35y\), and the side opposite \(\angle K\) is \(IJ\).

Step2: Determine the angle - side relationship

The angle opposite the shorter side is smaller. The side \(IK = 35y\) is the shortest side, so the angle opposite it (\(\angle J\)) is the smallest. The side \(IJ\) (using the fact that the sum of angles in a triangle is \(180^{\circ}\), \(\angle I+\angle J+\angle K=180^{\circ}\), \(\angle I = 180^{\circ}-\angle J - 95^{\circ}\)). Since \(JK>IK\), the angle opposite \(JK\) (\(\angle I\)) is larger than the angle opposite \(IK\) (\(\angle J\)). And \(\angle K = 95^{\circ}\) is the largest angle.

Answer:

\(m\angle J