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.

Explanation:

Step1: Recall tangent - radius property

A tangent to a circle is perpendicular to the radius at the point of tangency. Since \(PR\) is a tangent and \(XP\) is a radius (with \(X\) as the center and \(P\) the point of tangency), \(\angle XPR = 90^{\circ}\). \(\angle XPW\) is also \(90^{\circ}\) as it is a straight - line angle with \(\angle XPR\) and \(\angle XPW+\angle XPR = 180^{\circ}\).

Step2: Recall angle - in - a - semi - circle property

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

Answer:

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