Question

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

Step1: Identify side lengths

Sides: $AC=34v$, $BC=102v$, $AB$ (unknown, use triangle inequality)

Step2: Relate sides to angles

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

• Shortest side: $AC=34v$, opposite $\angle B$
• Longest side: $BC=102v$, opposite $\angle A$
• $\angle C=94^\circ$ is already given, which is obtuse (so larger than acute angles, but we confirm side order: $34v < AB < 102v$ since $102v - 34v < AB < 102v + 34v$, so $AB$ is between the two sides)

Step3: Order angles by side lengths

$\angle B$ (opposite shortest side) < $\angle A$ (opposite middle side) < $\angle C$ (opposite longest side, and obtuse)

Answer:

$m\angle B < m\angle A < m\angle C$