Question

In △ DEF, m∠ D=104° and m∠ E=25°. Which statement about the sides of △ DEF must be true?

a. DE>EF>FD
b. FD>DE>EF
c. EF>FD>DE
d. DE>FD>EF
e. EF>DE>FD
f. FD>EF>DE

Answer 1

To determine the relationship between the lengths of the sides in ΔDEF with angles measuring 104° and 25°, we find the third angle measures 51°. Based on angle sizes, the correct order of side lengths from longest to shortest is EF > DE > FD, which is option e.

Step-by-step explanation:

The question deals with the properties of triangles, specifically the relationship between the angles of a triangle and the lengths of its sides. To determine which statement about the sides of △DEF must be true when m∠ D=104° and m∠ E=25°, we use the fact that the sum of the angles in a triangle must be 180 degrees. Subtracting the measures of angles D and E from 180 degrees gives us the measure of angle F.

m∠ F = 180° - m∠ D - m∠ E

= 180° - 104° - 25°

= 51°

Next, we apply the principle that the longest side of a triangle is opposite the largest angle. In △DEF, angle D is the largest since 104° > 51° > 25°. Therefore, side EF opposite angle D must be the longest side. Angle E is the smallest, which means side FD opposite it is the shortest. This leads to the conclusion that EF > DE > FD, which corresponds to option e.