Question

list the angle measures of $\triangle pqr$ in order from smallest to largest. assume that $x$ is a positive number.
(there is a triangle $pqr$ with side $pr = 43x$, side $qr = 31x$, side $pq = 39x$)
$m\angle\square < m\angle\square < m\angle\square$

Explanation:

Step1: Match angles to opposite sides

In a triangle, an angle is opposite a side that does not share its vertex:

• $\angle P$ is opposite side $RQ = 31x$
• $\angle Q$ is opposite side $RP = 43x$
• $\angle R$ is opposite side $PQ = 39x$

Step2: Compare side lengths

Since $x>0$, compare coefficients:
$31x < 39x < 43x$

Step3: Relate sides to angles

By the triangle angle-side relationship: the smallest angle is opposite the shortest side, and the largest angle is opposite the longest side.
So the order of angles matches the reverse order of their opposite sides.

Answer:

$m\angle P < m\angle R < m\angle Q$