Question

1. ( m angle 2 = x + 146 )

Explanation:

Step1: Use the property of isosceles triangles

Since two sides of the triangle are equal (marked with the same tick), the base - angles of the isosceles triangle are equal. Let's assume the non - right - angle in the right - angled part of the figure. The sum of angles in a triangle is \(180^{\circ}\). In the right - angled triangle (with one angle \(90^{\circ}\)), if we consider the isosceles part, the non - right angle adjacent to the \(66^{\circ}\) angle is \(90^{\circ}-66^{\circ}=24^{\circ}\).

Step2: Use the exterior - angle property

The exterior angle (\(\angle2\)) of a triangle is equal to the sum of the two non - adjacent interior angles. The two non - adjacent interior angles here are \(90^{\circ}\) and \(24^{\circ}\). So \(m\angle2=90 + 24=114^{\circ}\).
Since \(m\angle2=x + 146\), we set up the equation \(x+146 = 114\).

Step3: Solve for \(x\)

Subtract 146 from both sides of the equation \(x+146=114\).
\(x=114 - 146=-32\)

Answer:

\(x=-32\)