Question

use the properties of exterior angles of triangles to find angle b.

Explanation:

Step1: Find angle \( a \)

Since \( a + 142^{\circ}=180^{\circ}\) (linear - pair of angles), then \( a = 180^{\circ}-142^{\circ}\).
\(a = 38^{\circ}\)

Step2: Use the angle - sum property of a right - triangle

In a right - triangle (one angle is \(90^{\circ}\)), the sum of angles is \(180^{\circ}\). Let the right angle be \(90^{\circ}\), angle \(a = 38^{\circ}\) and angle \(b\).
We know that \(a + b+90^{\circ}=180^{\circ}\). Substitute \(a = 38^{\circ}\) into the equation: \(38^{\circ}+b + 90^{\circ}=180^{\circ}\).
First, simplify the left - hand side: \(b+128^{\circ}=180^{\circ}\).
Then, solve for \(b\): \(b=180^{\circ}-128^{\circ}\).

Answer:

\(b = 52^{\circ}\)