Question

list the angle measures of $\triangle hij$ in order from smallest to largest. assume that $b$ is a positive number.

Explanation:

Step1: Recall the triangle angle - side relationship

In a triangle, the larger side is opposite the larger angle, and the smaller side is opposite the smaller angle. First, we need to compare the lengths of the sides of \(\triangle HIJ\). The sides are \(HI = 5b\), \(HJ=18b\), and we know that \(\angle H = 93^{\circ}\). Let's compare \(5b\) and \(18b\). Since \(b>0\), we can divide both sides of the inequality \(5b\) and \(18b\) by \(b\) (because \(b
eq0\) and \(b > 0\)) and get \(5<18\), so \(5b<18b\), which means \(HI < HJ\).

Step2: Determine the angles opposite the sides

In \(\triangle HIJ\), the side opposite \(\angle J\) is \(HI\), the side opposite \(\angle I\) is \(HJ\), and we know \(\angle H=93^{\circ}\). Since \(HI < HJ\), the angle opposite \(HI\) (\(\angle J\)) is smaller than the angle opposite \(HJ\) (\(\angle I\)). Also, the sum of the interior angles of a triangle is \(180^{\circ}\), so \(\angle I+\angle J+\angle H=180^{\circ}\), which means \(\angle I+\angle J=180 - 93=87^{\circ}\). So both \(\angle I\) and \(\angle J\) are less than \(93^{\circ}\), and \(\angle J<\angle I\) (because \(HI < HJ\)).

Step3: Order the angles

From the above analysis, we have \(\angle J<\angle I<\angle H\) (since \(\angle J\) is opposite the shortest side \(HI\), \(\angle I\) is opposite the longer side \(HJ\), and \(\angle H = 93^{\circ}\) is the largest angle as the sum of the other two angles is \(87^{\circ}\)).

Answer:

\(m\angle J