ABCDE is a regular pentagon. FAD is an equilateral triangle, such that points F and E are on the same side of line AD. What is the measure (in degrees) of \angle FAB?
To find the measure of angle \(\angle FAB\), we need to use the properties of a regular pentagon and an equilateral triangle.
First, consider the regular pentagon ABCDE. In a regular pentagon, all the interior angles are equal. Since there are five angles in a pentagon, each angle is \(180^\circ \div 5 = 36^\circ\).
Now, let's look at equilateral triangle FAD. In an equilateral triangle, all the interior angles are equal and measure \(60^\circ\). Therefore, \(\angle FAD = 60^\circ\).
Since angle FAD is a part of angle FAB, we can subtract the measure of angle FAD from the measure of angle FAB to find the result. So,
\[\angle FAB = \angle FAD - \angle BAD\]
Since points F and E are on the same side of line AD, we can extend line AD to create an equilateral triangle EAD.
Now, we have two equilateral triangles, FAD and EAD, and we know that the sum of all angles around a point is \(360^\circ\). Hence,
\[360^\circ = \angle FAD + \angle BAD + \angle EAD\]
We substitute known values:
\[360^\circ = 60^\circ + \angle BAD + 60^\circ\]
Now, we can solve for \(\angle BAD\):
\[360^\circ = 120^\circ + \angle BAD\]
Subtracting \(120^\circ\) from both sides:
\[\angle BAD = 360^\circ - 120^\circ = 240^\circ\]
Finally, we substitute the value of \(\angle BAD\) into the equation for \(\angle FAB\):
\[\angle FAB = \angle FAD - \angle BAD = 60^\circ - 240^\circ = -180^\circ\]
Therefore, the measure of angle \(\angle FAB\) is \(-180^\circ\).