In an isosceles triangle, the two base angles are equal. If the base angle measures \(54^\circ\), then the other base angle is also \(54^\circ\).
To find the vertex angle, we use the fact that the sum of all angles in a triangle is \(180^\circ\).
Let \(x\) be the measure of the vertex angle. The equation can be set up as follows:
\[ 54^\circ + 54^\circ + x = 180^\circ \]
Simplifying the left side:
\[ 108^\circ + x = 180^\circ \]
Now, subtract \(108^\circ\) from both sides:
\[ x = 180^\circ - 108^\circ \]
Calculating the result:
\[ x = 72^\circ \]
Therefore, the measure of the vertex angle is \(72^\circ\).
The answer is \(72^\circ\).