Label ΔABC.


The given information in ΔABC is

$\begin{array}{rcl}\angle A& =& 48°\\ \angle B& =& 80°\\ \angle C& =& 52°\\ b& =& 5.8 \mathrm{m}\end{array}$

$\angle A=48°$ and is across from side x. ($\angle A$ and side a are a matching pair.)
$\angle B=80°$ and is across from side b, which equals 5.8 m. ($\angle B$ and side b are a matching pair.)
$\angle C$ is not needed to find x.

$\begin{array}{rcl}\frac{a}{\sin A}& =& \frac{b}{\sin {\displaystyle B}}\\ \frac{a}{\sin {\displaystyle 48}{\displaystyle °}}& =& \frac{5.8}{\sin {\displaystyle 80}{\displaystyle °}}\\ \frac{a}{{\displaystyle \cancel{\sin 48°}}}\times \cancel{\sin 48°}& =& \frac{5.8}{\sin {\displaystyle 80}{\displaystyle °}}\times \sin 48°\\ a& =& \frac{5.8}{\sin {\displaystyle 80}{\displaystyle °}}\times \sin 48°\\ & =& 4.4 \mathrm{cm}\\ & & \end{array}$
Since x = a, the length of side x is 4.4 cm.

Label ΔABC.


The given information in ΔABC is

$\begin{array}{rcl}\angle A& =& 24°\\ \angle B& =& 15°\\ c& =& 91 \mathrm{in}\end{array}$

$\angle A=24°$ and is across from side a. ($\angle A$ and side a are a matching pair.)
$\angle B=15°$ and is across from side b or m. ($\angle B$ and side b are a matching pair.)
$\angle C$ is across from side c = 91 in. ($\angle C$ and side c are a matching pair.)

Since two angles are known, $\angle C$ can be calculated using the equation $\angle A+\angle B+\angle C=180°$.

$\begin{array}{rcl}\angle A+\angle B+\angle C& =& 180°\\ 24°+15°+\angle C& =& 180°\\ 39°+\angle C& =& 180°\\ \angle C& =& 180°-39°\\ & =& 141°\end{array}$

Substitute $\angle B, \angle C,$ and side c into the sine law and solve for side b.

$\begin{array}{rcl}\frac{b}{\sin B}& =& \frac{c}{\sin {\displaystyle C}}\\ \frac{b}{\sin {\displaystyle 15}{\displaystyle °}}& =& \frac{91}{\sin {\displaystyle 141}{\displaystyle °}}\\ \frac{b}{{\displaystyle \cancel{\sin 15°}}}\times \cancel{\sin 15°}& =& \frac{{\displaystyle 91}}{{\displaystyle \sin 141°}}\times \sin 15°\\ b& =& \frac{{\displaystyle 91}}{{\displaystyle \sin 141°}}\times \sin 15°\\ & =& 37.4 \mathrm{in}\end{array}$

Since b = m, the length of m is 37.4 in.