Question

in \\( \triangle def, m \angle d = 67 ^ { \circ } \\) and \\( m \angle e = 17 ^ { \circ } \\). which list has the sides of \\( \triangle def \\) in order from longest to shortest?
answer
\\( \bigcirc fd, de, ef \\)
\\( \bigcirc de, fd, ef \\)
\\( \bigcirc de, ef, fd \\)
\\( \bigcirc ef, fd, de \\)
\\( \bigcirc ef, de, fd \\)
\\( \bigcirc fd, ef, de \\)

Explanation:

Step1: Find the measure of angle \( F \)

The sum of angles in a triangle is \( 180^{\circ} \). So, \( m\angle F=180^{\circ}-m\angle D - m\angle E \).
Substitute \( m\angle D = 67^{\circ} \) and \( m\angle E = 17^{\circ} \):
\( m\angle F=180^{\circ}-67^{\circ}-17^{\circ}=96^{\circ} \).

Step2: Relate angles and sides

In a triangle, the larger the angle, the longer the side opposite it.
\( m\angle F = 96^{\circ}\), \(m\angle D=67^{\circ}\), \(m\angle E = 17^{\circ}\).
Side opposite \( \angle F\) is \( DE \), side opposite \( \angle D\) is \( EF \), side opposite \( \angle E\) is \( FD \).
Since \( 96^{\circ}>67^{\circ}>17^{\circ}\), then \( DE>EF>FD \).

Answer:

EF, DE, FD