Question

list the angle measures of $\triangle ghi$ in order from smallest to largest. assume that $a$ is a positive number.
$m\angle\square < m\angle\square < m\angle\square$

Explanation:

Step1: Compare side lengths

Since $a>0$, $41a < 144a$. The third side $GI$ can be inferred via triangle side-angle relationships, but we first use the rule: in a triangle, the smallest angle is opposite the shortest side, largest angle opposite the longest side.

Step2: Match angles to sides

• Side $GH=41a$ is opposite $\angle I$
• Side $HI=144a$ is opposite $\angle G$
• $\angle H=100^\circ$ is the largest angle (since the sum of triangle angles is $180^\circ$, the other two angles are acute, so smaller than $100^\circ$)

Step3: Order angles by size

$\angle I$ (opposite shortest side) < $\angle G$ (opposite longer side) < $\angle H$ (given $100^\circ$)

Answer:

$m\angle I < m\angle G < m\angle H$