Question

1. order the sides of $\triangle def$ from shortest to longest.

(image of triangle def with angles at e: 40°, at f: 75°; options: a. $\overline{de}$, b. $\overline{df}$, c. $\overline{ef}$)

Explanation:

Step1: Find angle at D

In a triangle, the sum of angles is \(180^\circ\). So, \(\angle D = 180^\circ - 40^\circ - 75^\circ = 65^\circ\).

Step2: Relate angles to sides

In a triangle, the larger the angle, the longer the side opposite to it.

• Opposite \(\angle E = 40^\circ\) is \(DF\) (side b).
• Opposite \(\angle D = 65^\circ\) is \(EF\) (side c).
• Opposite \(\angle F = 75^\circ\) is \(DE\) (side a).

Order of angles: \(40^\circ < 65^\circ < 75^\circ\) (i.e., \(\angle E < \angle D < \angle F\)).
Thus, order of sides (shortest to longest): \(DF < EF < DE\) (i.e., \(b < c < a\) or \(\overline{DF} < \overline{EF} < \overline{DE}\)).

Answer:

\(\overline{DF}\), \(\overline{EF}\), \(\overline{DE}\) (or \(b\), \(c\), \(a\))