1. **Stating the problem:** We have a quadrilateral inscribed in a circle with two known angles 70° and 80°, and two unknown angles labeled $\alpha$ and $\beta$. We need to find the values of $\alpha$ and $\beta$. 2. **Important rule:** The opposite angles of a cyclic quadrilateral (one inscribed in a circle) sum to 180°. This means: $$\alpha + 80^\circ = 180^\circ$$ and $$\beta + 70^\circ = 180^\circ$$ 3. **Calculate $\alpha$:** $$\alpha = 180^\circ - 80^\circ = 100^\circ$$ 4. **Calculate $\beta$:** $$\beta = 180^\circ - 70^\circ = 110^\circ$$ 5. **Conclusion:** The unknown angles are $\alpha = 100^\circ$ and $\beta = 110^\circ$.