To find angle \( b \), we can use the exterior angle theorem, which states that the exterior angle of a triangle is equal to the sum of the two non-adjacent interior angles.
Given:
- The exterior angle is \( 142^\circ \).
- The interior angles are \( a \) and \( b \).
According to the exterior angle theorem: \[ \text{Exterior Angle} = a + b \] So we have: \[ 142^\circ = a + b \]
We also know that in a right triangle, the sum of the two acute angles (in this case, \( a \) and \( b \)) is equal to \( 90^\circ \). Thus: \[ a + b = 90^\circ \]
Now, we have two equations:
- \( a + b = 142^\circ \) (from the exterior angle)
- \( a + b = 90^\circ \) (sum of angles in a triangle)
However, since the triangle is right-angled, we realize angle \( a \) and angle \( b \) must satisfy: \[ a + b + 90^\circ = 180^\circ \Rightarrow a + b = 90^\circ \] This suggests that \( a + b = 90^\circ\) is not appropriately used with the knowledge of the exterior angle \( 142^\circ \).
Since \( 142^\circ = 90^\circ + b \) (as the angle outside the triangle compensates for the right angle at point \( c \)), we can rearrange this to find \( b \): \[ b = 142^\circ - 90^\circ \]
Calculating: \[ b = 52^\circ \]
Therefore, the value of angle \( b \) is \( 52^\circ \).
The answer is: 52°
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