Since angles A and B are complementary, we have:
\[ m∠A + m∠B = 90° \]
We know that:
\[ m∠A = x° \] \[ m∠B = 2x° \]
Substituting the expressions into the equation gives:
\[ x + 2x = 90° \]
This simplifies to:
\[ 3x = 90° \]
To find \(x\), divide both sides by 3:
\[ x = 30° \]
Now we can find the measure of \(∠B\):
\[ m∠B = 2x = 2(30°) = 60° \]
Therefore, the measure of \(∠B\) is:
\[ \boxed{60°} \]