Question

given circle u with diameter vw and yxz tangent to u at point x. name an angle congruent to ∠vxw.

Explanation:

Step1: Use the tangent - radius property

Since \(YXZ\) is tangent to the circle \(U\) at point \(X\) and \(UX\) is the radius, \(UX\perp YXZ\). So, \(\angle VXW+\angle UXW = 90^{\circ}\). Also, because \(VW\) is a diameter, \(\angle VXW+\angle V = 90^{\circ}\) (angle in a semi - circle, \(\angle VXW\) is an inscribed angle subtended by the diameter \(VW\), so \(\angle VXW\) is a right angle. Wait, no, actually, since \(UX = UW\) (radii of the circle), \(\angle UWX=\angle UXW\).

Step2: Use angle - subtraction or angle - relationship

We know that \(\angle VXW\) and \(\angle VWX\) satisfy the following relationship. The tangent - radius theorem gives \(\angle VXW+\angle UXW = 90^{\circ}\). Since \(VW\) is a diameter, \(\angle VXW+\angle VWX=90^{\circ}\) (because \(\angle VXW\) is an inscribed angle subtended by the diameter \(VW\), \(\angle VXW = 90^{\circ}\) is wrong. Wait, correct approach: The measure of an angle between a tangent and a chord is equal to the measure of the inscribed angle subtended by the chord.
The angle between the tangent \(YXZ\) and the chord \(VX\) (\(\angle VXW\)) is equal to the inscribed angle subtended by the chord \(VX\). The inscribed angle subtended by chord \(VX\) is \(\angle VWX\)

Answer:

\(\angle VWX\)