In the diagram below, points A, E, and F lie on the same line. If ABCDE is a regular pentagon, and \angle EFD=90^\circ, then how many degrees are in the measure of \angle FDE?

The measure of ∠FDE = 18°

Explanation:

A Pentagon has 5 sides and is made of 3 triangles

So, sum of the interior angles of the triangle = 180°

Therefore, the total interior angle of a regular pentagon = 3 X 180° = 540°

A regular pentagon will have all its angle equal

All the five angles would make 540°

Let the measure of one angle = x

So,

5x = 540°

x = 108°

Therefore, the measure of each angle of a pentagon is 108°

From the diagram,

∠AED + ∠FED = 180°

∠AED = 108° as it is one of the sides of the pentagon

So,

108° + ∠FED = 180°

∠FED = 72°

In ΔEFD,

∠FED + ∠EFD + ∠FDE = 180°

72° + 90° + ∠FDE = 180°

∠FDE = 18°

Therefore, the measure of ∠FDE = 18°