Question

which statements about the diagram are true? select three options. □ de + ef > df □ △ def is an isosceles triangle. □ 5 < df < 13 □ de + df < ef □ △ def is a scalene triangle. (diagram: triangle def with e to d labeled 4, e to f labeled 9, and tick marks on sides de and df)

Explanation:

1. For \( DE + EF > DF \): By the triangle inequality theorem, the sum of two sides of a triangle is greater than the third side. Here, \( DE \) and \( EF \) are two sides, and \( DF \) is the third side, so \( DE + EF > DF \) holds.
2. For \( 5 < DF < 13 \): Using the triangle inequality theorem, the length of a side \( DF \) must satisfy \( |EF - DE| < DF < EF + DE \). Given \( DE = 4 \) and \( EF = 9 \), we have \( |9 - 4|=5 \) and \( 9 + 4 = 13 \), so \( 5 < DF < 13 \) is true.
3. For \( \triangle DEF \) being a scalene triangle: A scalene triangle has all sides of different lengths. We know \( DE = 4 \), \( EF = 9 \), and from the triangle inequality, \( DF \) is between 5 and 13 (not equal to 4 or 9, as \( 5

eq4,9 \) to be different from the other two sides), so all three sides are of different lengths, making it scalene.

The statement \( \triangle DEF \) is an isosceles triangle is false because an isosceles triangle has at least two equal sides, and here \( DE = 4 \), \( EF = 9 \), and \( DF \) is between 5 and 13 (not equal to 4 or 9). The statement \( DE + DF < EF \) is false by the triangle inequality theorem (sum of two sides should be greater than the third side, so \( DE + DF>EF \)).

Answer:

• \( DE + EF > DF \)
• \( 5 < DF < 13 \)
• \( \triangle DEF \) is a scalene triangle