Question

in \\( \triangle def, m\angle d = 117 ^ { \circ } \\) and \\( m\angle e = 26 ^ { \circ } \\). which list has the sides of \\( \triangle def \\) in order from shortest to longest?
answer
\\( \bigcirc ef, fd, de \\)
\\( \bigcirc ef, de, fd \\)
\\( \bigcirc fd, ef, de \\)
\\( \bigcirc fd, de, ef \\)
\\( \bigcirc de, fd, ef \\)
\\( \bigcirc de, ef, fd \\)

Explanation:

Step1: Find the measure of angle F

In a triangle, the sum of interior angles is \(180^\circ\). So, \(m\angle F = 180^\circ - m\angle D - m\angle E\).
Substitute \(m\angle D = 117^\circ\) and \(m\angle E = 26^\circ\) into the formula:
\(m\angle F = 180^\circ - 117^\circ - 26^\circ = 37^\circ\)

Step2: Order the angles from smallest to largest

Now we have the angles: \(m\angle E = 26^\circ\), \(m\angle F = 37^\circ\), \(m\angle D = 117^\circ\).
So, the order of angles from smallest to largest is \(\angle E < \angle F < \angle D\).

Step3: Relate angles to their opposite sides

In a triangle, the side opposite a smaller angle is shorter.

• The side opposite \(\angle E\) is \(FD\).
• The side opposite \(\angle F\) is \(DE\).
• The side opposite \(\angle D\) is \(EF\).

So, the order of sides from shortest to longest (opposite the order of angles from smallest to largest) is \(FD\), \(DE\), \(EF\).

Answer:

FD, DE, EF