1. The problem involves finding unknown angles $\theta$ in various circle diagrams where points lie on the circumference and $O$ is the center. 2. Key formulas and rules: - The angle at the center $O$ is twice the angle at the circumference subtended by the same chord: $$\angle AOB = 2 \times \angle ACB$$ - Angles in the same segment are equal. - The sum of angles in a triangle is $180^\circ$. - Opposite angles in cyclic quadrilaterals sum to $180^\circ$. 3. For each diagram, identify known angles and use these rules to set up equations involving $\theta$. 4. Example for diagram 1: - Given $\angle C = 140^\circ$ and $\theta$ at $A$. - Since $\angle C$ is an angle at the circumference subtended by chord $AB$, the central angle $\angle AOB = 2 \times 140^\circ = 280^\circ$ which is impossible (greater than $180^\circ$), so likely $\angle C$ is an exterior angle or another angle. - Instead, use triangle $ADB$ or cyclic quadrilateral properties to relate $\theta$ and $140^\circ$. 5. General approach: - Use the central angle theorem: $$\angle AOB = 2 \times \angle ACB$$ - Use triangle angle sum: $$\angle A + \angle B + \angle C = 180^\circ$$ - Use cyclic quadrilateral property: $$\angle A + \angle C = 180^\circ$$ 6. Solve the resulting equations step-by-step for $\theta$. 7. Repeat this process for each diagram using the given angles and the above rules. This method will allow you to find $\theta$ in each case by applying circle theorems and angle sum properties.