Question

use the figure below to answer the following questions. round your answers to two decimal places. enter deg after any degree values.
a) what is ( mangle e )?
b) what is ( ef )?
c) what is ( de )?

Explanation:

Step1: Recall triangle angle sum

In a right triangle, the sum of angles is \(180^\circ\), and one angle is \(90^\circ\). Given \(\angle D = 64^\circ\), \(\angle F = 90^\circ\). So \(\angle E=180^\circ - 90^\circ - 64^\circ\).
\(180 - 90 - 64 = 26\), so \(m\angle E = 26.00^\circ\) (rounded to two decimals).

Step2: Find \(EF\) (opposite to \(\angle D\), adjacent to \(\angle E\))

Using tangent: \(\tan(\angle D)=\frac{EF}{DF}\). \(DF = 9\), \(\angle D = 64^\circ\). So \(EF = DF\times\tan(64^\circ)\). \(\tan(64^\circ)\approx2.0503\), \(EF = 9\times2.0503\approx18.45\).

Step3: Find \(DE\) (hypotenuse)

Using cosine: \(\cos(\angle D)=\frac{DF}{DE}\). So \(DE=\frac{DF}{\cos(64^\circ)}\). \(\cos(64^\circ)\approx0.4384\), \(DE=\frac{9}{0.4384}\approx20.53\).

Answer:

s:
a) \(26.00^\circ\)
b) \(18.45\)
c) \(20.53\)