To find angle \( b \) in the right triangle using the properties of exterior angles, we start by noting two key pieces of information:
- The triangle has a right angle, which is \( 90^\circ \).
- The exterior angle at angle \( a \) is given as \( 142^\circ \).
According to the property of exterior angles for triangles, the exterior angle is equal to the sum of the two opposite interior angles.
In this case, the exterior angle \( 142^\circ \) can be expressed as:
\[
142^\circ = a + b
\]
Since we know this is a right triangle and one angle is a right angle (\( 90^\circ \)), the angles \( a \) and \( b \) must add up to \( 90^\circ \):
\[
a + b = 90^\circ
\]
To find angle \( b \), we can substitute \( a \) from one equation into the other. First, we can express \( a \) in terms of \( b \):
\[
a = 90^\circ - b
\]
Now we can substitute this expression for \( a \) into the equation for the exterior angle:
\[
142^\circ = (90^\circ - b) + b
\]
Simplifying gives:
\[
142^\circ = 90^\circ
\]
This is incorrect, so we should instead express \( a \) directly from the exterior angle equation:
Since we have:
\[
a + b = 142^\circ
\]
and
\[
a + b = 90^\circ
\]
To find both angles \( a \) and \( b \), let's resolve it.
We can break it down further: because \( 142^\circ \) is an exterior angle, it must also fulfill the relationship \( a + 90^\circ = 142^\circ \):
Thus:
\[
a = 142^\circ - 90^\circ
\]
Calculating gives:
\[
a = 52^\circ
\]
Now we can find \( b \):
Using:
\[
a + b = 90^\circ
\]
Substituting the known value of \( a \):
\[
52^\circ + b = 90^\circ
\]
Now, solving for \( b \):
\[
b = 90^\circ - 52^\circ = 38^\circ
\]
Thus, the value of angle \( b \) is
\[
\boxed{38^\circ}
\]
