Question

determine the unknown angle measures of △def. round to the nearest degree. m∠d = □° m∠f = □° (diagram: right triangle def with right angle at e, de = 23.5, ef = 24.3, df = 33.8)

Explanation:

Step1: Find \( m\angle D \) using sine

In right triangle \( DEF \), \( \sin(D)=\frac{EF}{DF} \). \( EF = 24.3 \), \( DF = 33.8 \).
\( \sin(D)=\frac{24.3}{33.8}\approx0.719 \)
\( m\angle D=\arcsin(0.719)\approx46^\circ \)

Step2: Find \( m\angle F \) using sine or angle sum

Using angle sum: \( m\angle F = 90^\circ - m\angle D \approx 90 - 46 = 44^\circ \) (or \( \sin(F)=\frac{DE}{DF}=\frac{23.5}{33.8}\approx0.695 \), \( \arcsin(0.695)\approx44^\circ \))

Answer:

\( m\angle D \approx 46^\circ \), \( m\angle F \approx 44^\circ \)