1. The problem asks to find the value of angle R in a regular polygon with five sides (a regular pentagon). 2. In a regular polygon, all interior angles are equal. The formula to find each interior angle of a regular polygon with $n$ sides is: $$\text{Interior angle} = \frac{(n-2) \times 180^\circ}{n}$$ 3. For a pentagon, $n=5$. Substitute this into the formula: $$\text{Interior angle} = \frac{(5-2) \times 180^\circ}{5} = \frac{3 \times 180^\circ}{5} = \frac{540^\circ}{5} = 108^\circ$$ 4. Therefore, each interior angle of a regular pentagon is $108^\circ$. 5. The angle R shown in the diagram is an exterior angle at a vertex of the pentagon. The exterior angle and interior angle at a vertex are supplementary, meaning they add up to $180^\circ$. 6. Calculate angle R: $$R = 180^\circ - 108^\circ = 72^\circ$$ 7. Hence, the value of angle R is $72^\circ$. This matches the expected exterior angle of a regular pentagon, which is $72^\circ$.