Question

determining unknown measures. what is the measure of angle e? m∠e = _. what is the length of ef rounded to the nearest hundredth? ef = _

Explanation:

Step1: Find angle E using angle - sum property of a triangle

The sum of angles in a triangle is 180°. So, $m\angle E=180^{\circ}-(43^{\circ} + 82^{\circ})$.
$m\angle E=180^{\circ}-125^{\circ}=55^{\circ}$

Step2: Use the Law of Sines to find EF

The Law of Sines states that $\frac{EF}{\sin D}=\frac{DF}{\sin E}$. We know that $DF = 15$, $D = 43^{\circ}$, and $E=55^{\circ}$. So, $EF=\frac{15\times\sin43^{\circ}}{\sin55^{\circ}}$.
$\sin43^{\circ}\approx0.682$, $\sin55^{\circ}\approx0.819$. Then $EF=\frac{15\times0.682}{0.819}=\frac{10.23}{0.819}\approx12.49$.

Answer:

$m\angle E = 55^{\circ}$
$EF\approx12.49$