Question

solve the triangle.
write each answer as an integer or as a decimal rounded to the nearest tenth.

Explanation:

Step1: Find angle at F

Sum of angles in triangle is $180^\circ$.
$\angle F = 180^\circ - 72^\circ - 84^\circ = 24^\circ$

Step2: Find side $e$ via Law of Sines

Relate side $e$ to known side/angle.
$\frac{e}{\sin 84^\circ} = \frac{12}{\sin 72^\circ}$
$e = \frac{12 \times \sin 84^\circ}{\sin 72^\circ}$
$e \approx \frac{12 \times 0.9945}{0.9511} \approx 12.5$

Step3: Find side $f$ via Law of Sines

Relate side $f$ to known side/angle.
$\frac{f}{\sin 24^\circ} = \frac{12}{\sin 72^\circ}$
$f = \frac{12 \times \sin 24^\circ}{\sin 72^\circ}$
$f \approx \frac{12 \times 0.4067}{0.9511} \approx 5.1$

Answer:

$\angle F = 24^\circ$, $e \approx 12.5$, $f \approx 5.1$