In a circle, the inscribed angle \( C \) that subtends an arc (in this case, the arc \( AB \)) is half the measure of the angle formed at the center of the circle that subtends the same arc. However, we can also use the property that the sum of the angles in triangle \( ABC \) is always \( 180^\circ \).
Given:
- \( m\angle A = 27^\circ \)
- \( m\angle C \) is an inscribed angle that subtends the diameter \( AB \), hence \( m\angle C = 90^\circ \) (because an inscribed angle that subtends a semicircle is always \( 90^\circ \)).
To find \( m\angle B \), we can use the fact that the sum of the angles in triangle \( ABC \) is \( 180^\circ \):
\[ m\angle A + m\angle B + m\angle C = 180^\circ \]
Substituting the known measures:
\[ 27^\circ + m\angle B + 90^\circ = 180^\circ \]
Now, combine the known angles:
\[ 117^\circ + m\angle B = 180^\circ \]
Subtracting \( 117^\circ \) from both sides gives:
\[ m\angle B = 180^\circ - 117^\circ = 63^\circ \]
Therefore, the measure of angle \( B \) is:
\[ \boxed{63^\circ} \]
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