Question

list the angle measures of \\(\triangle tuv\\) in order from smallest to largest. assume that \\(p\\) is a positive number.
\\(m\angle\square < m\angle\square < m\angle\square\\)

Explanation:

Step1: Relate sides to opposite angles

In a triangle, the length of a side is proportional to the measure of its opposite angle: shorter side → smaller opposite angle, longer side → larger opposite angle.
Opposite angles:

• Side $43p$ → $\angle T$
• Side $138p$ → $\angle U$
• Side $UT$ (opposite $\angle V$)

Step2: Compare side lengths

Since $p>0$, $43p < 138p$. Also, by triangle inequality, $UT < 43p + 138p = 181p$, and $UT > 138p - 43p = 95p$, so $43p < 138p < UT$.

Step3: Match to angle measures

From side order $43p < 138p < UT$, the opposite angles follow: $\angle T < \angle U < \angle V$. We confirm $\angle V=93^\circ$, which is the largest, consistent with the longest side opposite it.

Answer:

$m\angle T < m\angle U < m\angle V$