1. **State the problem:** We need to find the measure of each interior angle of a regular octagon. 2. **Formula for the sum of interior angles:** The sum of interior angles of a polygon with $n$ sides is given by: $$\text{Sum} = (n - 2) \times 180$$ 3. **Calculate the sum for an octagon:** Since an octagon has $n = 8$ sides, $$\text{Sum} = (8 - 2) \times 180 = 6 \times 180 = 1080$$ 4. **Find each interior angle:** For a regular polygon, all interior angles are equal, so each interior angle is: $$\text{Each angle} = \frac{\text{Sum}}{n} = \frac{1080}{8} = 135$$ 5. **Conclusion:** Each interior angle of a regular octagon measures $135$ degrees.