Question

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

Explanation:

Step1: Find angle E

The sum of angles in a triangle is $180^\circ$.
$\angle E = 180^\circ - 19^\circ - 44^\circ = 117^\circ$

Step2: Calculate side $d$ via Law of Sines

Relate $d$, $\angle D$, $DF$, $\angle E$.
$\frac{d}{\sin(19^\circ)} = \frac{14}{\sin(117^\circ)}$
$d = \frac{14 \times \sin(19^\circ)}{\sin(117^\circ)}$
$d \approx \frac{14 \times 0.3256}{0.8910} \approx 5.1$

Step3: Calculate side $f$ via Law of Sines

Relate $f$, $\angle F$, $DF$, $\angle E$.
$\frac{f}{\sin(44^\circ)} = \frac{14}{\sin(117^\circ)}$
$f = \frac{14 \times \sin(44^\circ)}{\sin(117^\circ)}$
$f \approx \frac{14 \times 0.6947}{0.8910} \approx 10.9$

Answer:

$\angle E = 117^\circ$, $d \approx 5.1$, $f \approx 10.9$