To find angle \( b \) in the right triangle, we can utilize the property that the exterior angle is equal to the sum of the two remote interior angles. In this case, the exterior angle is given as \( 142^\circ \), and the two remote interior angles are \( a \) and \( b \).
We know that the angles in a triangle add up to \( 180^\circ \):
- In a right triangle, one angle is \( 90^\circ \) (the right angle).
- Therefore, the sum of angles \( a \) and \( b \) in the triangle is \( 90^\circ \) (since \( 90^\circ + a + b = 180^\circ \)).
According to the exterior angle theorem: \[ \text{Exterior angle} = \text{angle } a + \text{angle } b \]
Thus, we can write: \[ 142^\circ = a + b \]
Combining the two equations we have:
- \( a + b = 90^\circ \)
- \( a + b = 142^\circ \)
However, since \( a \) and \( b \) must add to \( 90^\circ \) due to the right triangle property, we realize that the exterior angle \( 142^\circ \) is not involving the right angle directly but is related to the triangle interior angles.
We can express angle a using the first equation: \[ a = 90^\circ - b \] Substituting this into the exterior angle equation: \[ 142^\circ = (90^\circ - b) + b \] This simplifies to: \[ 142^\circ = 90^\circ \] This leads to an inconsistent statement, showing that the angles \( a \) and \( b \) composition must lead us to simply subtract from \( 142^\circ \).
Subtracting the sum of the known angle: \[ b = 142^\circ - 90^\circ \] \[ b = 52^\circ \]
Thus, the measure of angle \( b \) is: \[ \boxed{52^\circ} \]
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