1. **Problem statement:** We are given a circle with center $O$ and points $P$ and $Q$ on the circumference. The central angle $\angle POQ$ is $280^\circ$. We need to find the angle $x$ at point $R$ formed by lines $PR$ and $QR$ outside the circle. 2. **Relevant theorem:** The angle formed outside the circle by two secants (or tangents) intersecting at a point outside the circle is half the difference of the measures of the intercepted arcs. 3. **Step 1: Identify arcs:** The central angle $\angle POQ = 280^\circ$ intercepts the major arc $PQ$ of $280^\circ$. The minor arc $PQ$ is the remainder of the circle: $$360^\circ - 280^\circ = 80^\circ$$ 4. **Step 2: Apply the external angle formula:** The angle $x$ at $R$ is half the difference of the intercepted arcs: $$x = \frac{|\text{major arc} - \text{minor arc}|}{2} = \frac{|280^\circ - 80^\circ|}{2}$$ 5. **Step 3: Calculate:** $$x = \frac{200^\circ}{2} = 100^\circ$$ 6. **Answer:** The angle $x$ at point $R$ is $100^\circ$. This uses the property that the angle formed outside a circle by two secants is half the difference of the intercepted arcs.