Since angles \( \angle A \) and \( \angle B \) are complementary, we have:
\[ m\angle A + m\angle B = 90^\circ \]
Given that \( m\angle A = x^\circ \) and \( m\angle B = 2x^\circ \), we can substitute these values into the equation:
\[ x + 2x = 90^\circ \]
Combining like terms gives:
\[ 3x = 90^\circ \]
Now, solving for \( x \):
\[ x = \frac{90^\circ}{3} = 30^\circ \]
Now we can find \( m\angle B \):
\[ m\angle B = 2x = 2 \times 30^\circ = 60^\circ \]
Therefore, the measure of \( \angle B \) is \( \boxed{60^\circ} \).
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