Question

1. find ( mangle v ).

Explanation:

Step1: Use the Angle - Angle (AA) similarity criterion

Since \(\angle U=\angle R = 90^{\circ}\) and the two triangles are similar (by AA similarity as the non - right angles are corresponding). Corresponding angles of similar triangles are equal.

Step2: Apply the triangle angle - sum property

In \(\triangle QRS\), we know that the sum of angles in a triangle is \(180^{\circ}\). Let \(\angle Q + \angle R+\angle S=180^{\circ}\). Since \(\angle R = 90^{\circ}\) and \(\angle S=74^{\circ}\), then \(\angle Q=180^{\circ}-(90^{\circ} + 74^{\circ})=16^{\circ}\).
In \(\triangle TUV\), \(\angle U = 90^{\circ}\), and using the angle - sum property \(\angle T+\angle U+\angle V=180^{\circ}\). Also, \(\angle T\) corresponds to \(\angle Q\) and \(\angle V\) corresponds to \(\angle S\) (by similarity of triangles).

Another way:
Since the two right - angled triangles are similar (by AA, as they are both right - angled and the non - right angles are in proportion).
In a right - angled triangle, if we consider \(\triangle TUV\) and \(\triangle QRS\), \(\angle V\) and \(\angle S\) are corresponding angles.

Answer:

\(m\angle V = 16^{\circ}\)