Question

determining unknown measures
what is the measure of angle e?
m∠e = 55
what is the length of $overline{ef}$ rounded to the nearest hundredth?
ef ≈ dropdown with 10.33, 12.49, 16.26, 18.13
(triangle with vertices d, e, f: angle at d = 43°, angle at f = 82°, side df = 15)

Explanation:

Step1: Find angle E

The sum of angles in a triangle is \(180^\circ\). So, \(m\angle E = 180^\circ - 43^\circ - 82^\circ = 55^\circ\) (which matches the given value).

Step2: Apply the Law of Sines

The Law of Sines states that \(\frac{EF}{\sin D}=\frac{DF}{\sin E}\). We know \(DF = 15\), \(m\angle D = 43^\circ\), \(m\angle E = 55^\circ\). So, \(EF=\frac{DF\times\sin D}{\sin E}\).

Step3: Calculate the values

First, find \(\sin 43^\circ\approx0.6820\) and \(\sin 55^\circ\approx0.8192\). Then, \(EF=\frac{15\times0.6820}{0.8192}=\frac{10.23}{0.8192}\approx12.49\).

Answer:

The length of \(\overline{EF}\) is \(12.49\).