Question

determining unknown measures
what is the measure of angle e?
$m\angle e = \quad ^\circ$
what is the length of $\overline{ef}$ rounded to the nearest hundredth?
$ef \approx$

Explanation:

Step1: Calculate angle E

The sum of angles in a triangle is $180^\circ$. Subtract the known angles from $180^\circ$.
$m\angle E = 180^\circ - 43^\circ - 82^\circ = 55^\circ$

Step2: Apply Law of Sines for EF

Set up the proportion using Law of Sines: $\frac{EF}{\sin(D)} = \frac{DF}{\sin(E)}$.
Substitute values: $EF = \frac{DF \cdot \sin(D)}{\sin(E)} = \frac{15 \cdot \sin(43^\circ)}{\sin(55^\circ)}$

Step3: Compute the length

Calculate the sines and solve: $\sin(43^\circ) \approx 0.6820$, $\sin(55^\circ) \approx 0.8192$
$EF \approx \frac{15 \times 0.6820}{0.8192} \approx 12.53$

Answer:

$m\angle E = 55^\circ$
$EF \approx 12.53$