Question

what is the measure of circumscribed ∠x? 45° 50° 90° 95°

Explanation:

Step1: Recall the property of a quadrilateral circumscribed about a circle

The sum of the measures of opposite angles in a quadrilateral circumscribed about a circle is \(180^{\circ}\). Also, the angle between a tangent and a radius is \(90^{\circ}\). In quadrilateral \(VYXW\), \(\angle VYX = 90^{\circ}\) (tangent - radius property) and \(\angle VWX=90^{\circ}\) (tangent - radius property).

Step2: Use the angle - sum property of a quadrilateral

The sum of the interior angles of a quadrilateral is \(360^{\circ}\). Let \(\angle X\) be \(x\), \(\angle V = 130^{\circ}\) (given the right - angle parts, assume the central angle \( \angle V\) is calculated from the figure's context, but more accurately, using the formula \(x+\angle V + 90^{\circ}+90^{\circ}=360^{\circ}\). If we assume the central angle \(\angle V\) is \(130^{\circ}\) (by the property that the sum of angles in a quadrilateral \(VYXW\): \(x + 90^{\circ}+90^{\circ}+\angle V=360^{\circ}\), and if we know from the circle - tangent - radius relations and the fact that the sum of non - \(x\) and non - \(90^{\circ}\) angles: \(\angle V\) and \(x\) are supplementary to the sum of the two right angles. Another way: The measure of the angle between two tangents (\(\angle X\)) and the central angle (\(\angle V\)) satisfy \(\angle X+\angle V = 180^{\circ}\). If \(\angle V = 130^{\circ}\) (from the fact that the sum of angles in the quadrilateral \(VYXW\): \(360-(90 + 90)=180\) for \(\angle X+\angle V\), and if we assume the central angle is \(130^{\circ}\) (by the relation of the circle's arc and the angles formed by tangents and radii), then \(x=50^{\circ}\))

Answer:

\(50^{\circ}\)