To find the measure of angle \( B \) in triangle \( ABC \), we can use the fact that the sum of the angles in a triangle is \( 180^\circ \).
We have:
- \( m\angle A = 63^\circ \)
- \( m\angle C = 99^\circ \)
Using the triangle angle sum property:
\[ m\angle A + m\angle B + m\angle C = 180^\circ \]
Substituting the values we know:
\[ 63^\circ + m\angle B + 99^\circ = 180^\circ \]
Combining \( 63^\circ \) and \( 99^\circ \):
\[ 162^\circ + m\angle B = 180^\circ \]
Now, to find \( m\angle B \):
\[ m\angle B = 180^\circ - 162^\circ \]
\[ m\angle B = 18^\circ \]
Thus, the measure of angle \( B \) is \( 18^\circ \).