Question

determine the unknown angle measures of △def. round to the nearest degree. m∠d = \\(\square\\)° m∠f = \\(\square\\)°

Explanation:

Step1: Find \( m\angle D \)

We can use the cosine function for \( \angle D \) in right triangle \( DEF \). The adjacent side to \( \angle D \) is \( DE = 23.5 \) and the hypotenuse is \( DF = 33.8 \). So, \( \cos(D)=\frac{DE}{DF}=\frac{23.5}{33.8} \).
\[
\cos(D)=\frac{23.5}{33.8}\approx0.6953
\]
Then, \( \angle D=\arccos(0.6953) \). Using a calculator, \( \angle D\approx46^\circ \) (rounded to the nearest degree).

Step2: Find \( m\angle F \)

We know that the sum of angles in a triangle is \( 180^\circ \), and \( \angle E = 90^\circ \). So, \( \angle F=90^\circ - \angle D \).
\[
\angle F = 90^\circ - 46^\circ = 44^\circ
\]
(We can also verify using sine for \( \angle F \): \( \sin(F)=\frac{DE}{DF}=\frac{23.5}{33.8}\approx0.6953 \), \( \angle F=\arcsin(0.6953)\approx44^\circ \))

Answer:

\( m\angle D = \boxed{46}^\circ \)
\( m\angle F = \boxed{44}^\circ \)