Question

the figure below shows a circle with center x, diameter pd, and tangent pr. which of the angles must be right angles? select all that apply.
answer
∠upv
∠pud
∠vpr
∠upx
∠xpw

Explanation:

Step1: Recall tangent - radius property

A tangent to a circle is perpendicular to the radius at the point of tangency. Since \(PR\) is tangent to the circle with center \(X\) at point \(P\), \(\angle XPR = 90^{\circ}\). Also, \(\angle XPW= 180^{\circ}-\angle XPR = 90^{\circ}\)

Step2: Recall the inscribed - angle property

An angle inscribed in a semi - circle is a right angle. Since \(PD\) is a diameter of the circle, \(\angle PUD = 90^{\circ}\)

Answer:

\(\angle PUD\), \(\angle XPW\)