To find the measure of a base angle of one of the triangles, we first need to determine the central angle formed by the vertices of the pentagon at the center of the shape.
- The total degrees around a point (the center of the pentagon) is 360°.
- Since a regular pentagon has 5 equal central angles, we can calculate the measure of each central angle:
\[ \text{Central angle} = \frac{360°}{5} = 72° \]
- Each of the isosceles triangles has its vertex at the center of the pentagon and its two other vertices on the pentagon at equal distances from each other. Therefore, the vertex angle of each isosceles triangle (which is also the central angle) is 72°.
- We now need to find the base angles of one of these isosceles triangles. The sum of angles in a triangle is 180°, and since the triangle is isosceles, the two base angles are equal. Let the measure of each base angle be \( b \). Then we have:
\[ 72° + 2b = 180° \]
- Solving for \( b \):
\[ 2b = 180° - 72° \] \[ 2b = 108° \] \[ b = \frac{108°}{2} = 54° \]
Thus, the measure of a base angle of one of the triangles is \( \boxed{54°} \).