Question

1. ( mangle 2=x + 83 )

Explanation:

Step1: Find the adjacent angle

The adjacent angle of \(142^{\circ}\) is \(180 - 142=38^{\circ}\).

Step2: Use the property of the triangle

Since the two - segment are equal, the triangle is isosceles. And we know one angle is \(38^{\circ}\), and there is a right - angle (\(90^{\circ}\)). Let's use the angle sum property of a triangle. But another way: The exterior angle property (not the main triangle). Wait, more accurately, using the fact that in the isosceles triangle part. The angle adjacent to \(142^{\circ}\) is \(38^{\circ}\). And since the two sides are equal (marked with the same tick), the base angles of the small isosceles triangle: \(m\angle2-(90^{\circ}-38^{\circ}) = 38^{\circ}\) (using the property that the sum of angles in a right - angled part). So \(m\angle2= 65^{\circ}\). Then \(x + 83=65\), \(x=65 - 83=- 18\).

Answer:

\(x=-18\)