In the adjoining figure, AOB is a diameter of the circle with center O. QR is a chord and the tangents to the circle at points Q and R include an angle of measure $$72^\circ$$. Chords BQ and AR intersect at P, then the measure of $$\angle RPB$$ is

The radii to Q and R are perpendicular to the two tangents, so the angle between the tangents and the central angle are supplementary, giving $$\angle QOR = 180^\circ - 72^\circ = 108^\circ$$, so arc QR is $$108^\circ$$. Q and R lie on the semicircle from A to B, so arc AQ plus arc RB $$= 180^\circ - 108^\circ = 72^\circ$$. The angle between the two intersecting chords equals half the sum of the intercepted arcs, so $$\angle RPB = \frac{72^\circ}{2} = 36^\circ$$.