1. **Problem statement:** Find the value of $x$ in each of the six circles based on the given angles. 2. **Important rules and formulas:** - The sum of angles around a point is $360^\circ$. - The angle subtended by a diameter in a circle is $90^\circ$. - The exterior angle of a triangle equals the sum of the two opposite interior angles. - Angles in the same segment of a circle are equal. - Tangent and radius form a $90^\circ$ angle. --- ### Top-left circle: 3. Given angles: $31^\circ$ (left), $101^\circ$ (interior), find $x$ (right). 4. The sum of angles on a straight line (diameter) is $180^\circ$. 5. Calculate $x$: $$x = 180^\circ - 101^\circ = 79^\circ$$ --- ### Top-middle circle: 6. Given angle: $64^\circ$ (left), find $x$ (upper-right). 7. Angles subtended by the same chord are equal, so $x = 64^\circ$. --- ### Top-right circle: 8. Given angles: $73^\circ$ (upper-left), $60^\circ$ (top), find $x$ (right). 9. Sum of angles in triangle is $180^\circ$: $$x = 180^\circ - 73^\circ - 60^\circ = 47^\circ$$ --- ### Bottom-left circle: 10. Given tangent angle $73^\circ$ (left-bottom), interior angle $60^\circ$ (bottom), find $x$ (upper-left). 11. Tangent and radius form $90^\circ$, so the angle adjacent to $73^\circ$ is: $$90^\circ - 73^\circ = 17^\circ$$ 12. Sum of angles in triangle: $$x = 180^\circ - 60^\circ - 17^\circ = 103^\circ$$ --- ### Bottom-middle circle: 13. Given angles: $126^\circ$ (left), $54^\circ$ (right), find $x$ (upper-left). 14. Sum of angles around a point is $360^\circ$: $$x = 360^\circ - 126^\circ - 54^\circ = 180^\circ$$ --- ### Bottom-right circle: 15. Given exterior angle $x$ (left), interior angle $40^\circ$ (right), tangent lines at lower-left and upper-left. 16. Exterior angle equals sum of opposite interior angles: $$x = 40^\circ + 90^\circ = 130^\circ$$ (The $90^\circ$ comes from tangent-radius angle.) --- **Final answers:** - Top-left: $x = 79^\circ$ - Top-middle: $x = 64^\circ$ - Top-right: $x = 47^\circ$ - Bottom-left: $x = 103^\circ$ - Bottom-middle: $x = 180^\circ$ - Bottom-right: $x = 130^\circ$