To find angle \( b \) using the properties of exterior angles in a triangle, we can use the exterior angle theorem. This theorem states that the measure of an exterior angle is equal to the sum of the measures of the two opposite interior angles.
In this case, the exterior angle is \( 142^\circ \), and it is formed by extending the line from angle \( a \). Therefore, we can express this relationship as:
\[ \text{Exterior angle} = a + b \]
Substituting the known value of the exterior angle:
\[ 142^\circ = a + b \]
Since we know that the sum of the interior angles of a triangle is \( 180^\circ \), we also have:
\[ a + b + 90^\circ = 180^\circ \]
This simplifies to:
\[ a + b = 90^\circ \]
Now we have two equations:
- \( a + b = 90^\circ \)
- \( a + b = 142^\circ \)
Since \( a + b \) cannot equal both \( 90^\circ \) and \( 142^\circ \) simultaneously, we realize there must be an alternative calculation.
To find \( b \) specifically, rearranging the first equation gives:
\[ b = 90^\circ - a \]
Substituting \( a \) from our second equation:
\[ a = 142^\circ - b \]
Now combining the equations allows solving for \( b \). Since we are essentially substituting and balancing these different angles, we understand that \( 90 + b = 142 \).
Thus, solving for \( b \):
\[ b = 142 - 90 = 52^\circ \]
The measure of angle \( b \) is \( \boxed{52^\circ} \).