Question

list the side lengths of $\triangle ghi$ in order from shortest to longest, given that $m\angle h = 33^\circ$, $m\angle i = 84^\circ$, and $m\angle g = 63^\circ$.
$\square < \square < \square$
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Explanation:

Step1: Recall Angle-Side Relationship

In a triangle, the longest side is opposite the largest angle, and the shortest side is opposite the smallest angle.

Step2: Order the Angles

Given \( m\angle H = 33^\circ \), \( m\angle G = 63^\circ \), \( m\angle I = 84^\circ \). Order angles from smallest to largest: \( \angle H < \angle G < \angle I \).

Step3: Determine Opposite Sides

• Side opposite \( \angle H \) is \( GI \).
• Side opposite \( \angle G \) is \( HI \).
• Side opposite \( \angle I \) is \( GH \).

Since angle order is \( \angle H < \angle G < \angle I \), side order (opposite) is \( GI < HI < GH \).

Answer:

\( GI < HI < GH \)