The measure of an angle inscribed in a circle is half the measure of the central angle that intercepts the same arc. That is,
mDA= Also, the sum of the interior angles of quadrilateral ABCP is
360°, and the measure of the obtuse angle P is
360°−x°.Hence,
.simplifying this equation gives
=40°, and so
x=80Alternate approach: If points A and P are joined, then the triangles that will be formed, APB and APC, are isosceles because
PA=PB=PC. It follows that the base angles on both triangles each have measure of
20°. Angle A consists of two base angles, and therefore,
m∠A=40°.Since the measure of an angle inscribed in a circle is half the measure of the central angle that intercepts the same arc, it follows that the value of
x is
80°.