Answer
$m \angle R = 82^{\circ}$
Work Step by Step
Let's set up the law of cosines to find $m \angle B$:
$b^2 = a^2 + c^2 - 2ac$ cos $\angle B$
In this exercise, $a = \overline{RS}$, $b = \overline{QS}$, and $c = \overline{QR}$. We want to find the measure of $\angle R$, which is $\angle B$ in the formula.
Let's plug in what we know:
$37.6^2 = 25.2^2 + 31.9^2 - 2(31.9)(25.2)$ cos $m \angle R$
Evaluate exponents first, according to order of operations:
$1413.76 = 635.04 + 1017.61 - 2(31.9)(25.2)$ cos $m \angle R$
Add to simplify on the right side of the equation:
$1413.76 = 1652.65 - 2(31.9)(25.2)$ cos $m \angle R$
Multiply to simplify:
$1413.76 = 1652.65 - 1607.76$ cos $m \angle R$
Subtract $1652.65$ from each side of the equation to move constants to the left side of the equation:
$-238.89 = -1607.76$ cos $m \angle R$
Divide each side by $-1607.76$:
cos $m \angle R$ = $\frac{-238.89}{-1607.76}$
Take $cos^{-1}$ to solve for $\angle R$:
$m \angle R = 82^{\circ}$
