Question

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list the angle measures of △ghi in order from smallest to largest. assume that b is a positive number.

g
169b
92°
h
49b
i
m∠□ < m∠□ < m∠□

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Explanation:

Step1: Analyze side lengths

In a triangle, the larger the side length, the larger the opposite angle. The sides are \(49b\) (opposite \(\angle H\)), \(169b\) (opposite \(\angle I\)), and we know \(\angle G = 92^\circ\). Compare \(49b\) and \(169b\): since \(b>0\), \(49b < 169b\). So the side opposite \(\angle H\) (\(49b\)) is shorter than the side opposite \(\angle I\) (\(169b\)), meaning \(\angle H < \angle I\). Also, \(\angle G = 92^\circ\), which is an obtuse angle, so it's the largest angle in the triangle (since the sum of angles in a triangle is \(180^\circ\), and an obtuse angle is greater than \(90^\circ\), so the other two angles must be acute and sum to \(180 - 92 = 88^\circ\), so both \(\angle H\) and \(\angle I\) are less than \(92^\circ\)).

Step2: Order the angles

From the side - angle relationship: side \(49b\) (opposite \(\angle H\)) < side \(169b\) (opposite \(\angle I\)) < side opposite \(\angle G\) (but \(\angle G\) is already known as \(92^\circ\), and \(\angle H\) and \(\angle I\) are acute). So the order of angles from smallest to largest is \(\angle H\), \(\angle I\), \(\angle G\).

Answer:

\(m\angle H < m\angle I < m\angle G\)