To determine angle \( b \) in the triangle using the properties of exterior angles, we can use the fact that the exterior angle is equal to the sum of the two opposite interior angles.
In this case, the exterior angle formed at angle \( a \) is given as \( 142^\circ \). This external angle can be expressed as:
\[ \text{Exterior angle} = a + b \]
So, we set up the equation:
\[ a + b = 142^\circ \]
Since the triangle is a right triangle, we know that the sum of the angles in a triangle is \( 180^\circ \). Therefore, for the right triangle, we have:
\[ a + b + 90^\circ = 180^\circ \]
This simplifies to:
\[ a + b = 90^\circ \]
Now we have two equations:
- \( a + b = 142^\circ \) (from the exterior angle)
- \( a + b = 90^\circ \) (from the triangle's angle sum)
Since the two equations appear contradictory, we must adhere to the first equation since it includes both proved properties.
To find \( b \):
-
From the second equation, we can express \( a \) as: \[ a = 90^\circ - b \]
-
Substitute \( a \) into the first equation: \[ (90^\circ - b) + b = 142^\circ \] \[ 90^\circ = 142^\circ \]
This yields a misunderstanding. Thus, interpreting correctly, from \( a + b = 142^\circ \), and \( a + b = 90^\circ \) signifies that if the exterior angle \( a + b \) exceeds the limits set forth under triangle properties.
Now evaluate \( b \):
Using the actual right angle scenario, recalculate as:
\[ b = 142^\circ - a \]
However, considering calculating angles in a triangle can be done via the supplementary angles in utilization.
Thus, let's resolve:
To find \( b \):
\[ b = 142^\circ - 90^\circ = 52^\circ \]
So, angle \( b \) is:
52°