1. The problem asks to find the measures of angles $\alpha$ and $\beta$ in a regular octagon, where the polygon has 8 equal sides and equal interior angles. 2. First, recall the formula for the measure of each interior angle in a regular polygon with $n$ sides: $$\text{Interior angle} = \frac{(n-2) \times 180^\circ}{n}$$ 3. For an octagon, $n=8$, so each interior angle is: $$\frac{(8-2) \times 180^\circ}{8} = \frac{6 \times 180^\circ}{8} = \frac{1080^\circ}{8} = 135^\circ$$ 4. Since the polygon is regular, all interior angles are $135^\circ$. 5. The diagonal divides the octagon into two parts, creating angles $\alpha$ and $\beta$ at vertices connected by the diagonal. 6. To find $\alpha$, note that it is an interior angle of the polygon, so: $$\alpha = 135^\circ$$ 7. To find $\beta$, observe that it is the angle formed between the diagonal and one side of the polygon at a vertex. 8. The diagonal in a regular octagon creates isosceles triangles. The central angle between two adjacent vertices is: $$\frac{360^\circ}{8} = 45^\circ$$ 9. The diagonal connects vertices that are 3 edges apart, so the central angle subtended by this diagonal is: $$3 \times 45^\circ = 135^\circ$$ 10. The triangle formed by two radii and the diagonal has two equal sides (radii), so the base angles are equal. 11. The sum of angles in this triangle is $180^\circ$, so each base angle is: $$\frac{180^\circ - 135^\circ}{2} = \frac{45^\circ}{2} = 22.5^\circ$$ 12. The angle $\beta$ is the exterior angle adjacent to one of these base angles, so: $$\beta = 180^\circ - 22.5^\circ = 157.5^\circ$$ 13. Rounding to the nearest degree: $$\alpha = 135^\circ$$ $$\beta = 158^\circ$$ Final answers: $$\alpha = 135^\circ$$ $$\beta = 158^\circ$$