1. **Problem Statement:** Find the values of angles $x$ and $y$ in the given circle diagrams using circle theorems and angle properties. 2. **Key Theorems and Rules:** - **Inscribed Angle Theorem:** An angle inscribed in a circle is half the measure of its intercepted arc. - **Angles in the Same Segment:** Angles subtended by the same chord and on the same side of the chord are equal. - **Opposite Angles of Cyclic Quadrilateral:** Sum to 180°. - **Vertical Angles:** Opposite angles formed by two intersecting lines are equal. --- ### Part a (Graph a): 3. Given: - Angle at $D$ between chords $CD$ and $FD$ is $74^\circ$. - Angle at $D$ between chords $BD$ and $FD$ is $28^\circ$. - Angles $x$ at $C$ (two marked equal angles). - Angle $y$ near $B$. 4. Since $x$ is marked twice at $C$, these two angles are equal. 5. The angle at $D$ between $CD$ and $FD$ is $74^\circ$, and between $BD$ and $FD$ is $28^\circ$. 6. The angle between $BD$ and $CD$ at $D$ is $74^\circ - 28^\circ = 46^\circ$. 7. By the Inscribed Angle Theorem, angle $x$ at $C$ subtends the same arc as angle at $D$ between $BD$ and $CD$, so: $$x = \frac{1}{2} \times 46^\circ = 23^\circ$$ 8. Angle $y$ near $B$ is an inscribed angle subtending the same arc as angle $x$ at $C$, so $y = x = 23^\circ$. --- ### Part b (Graph b): 9. Given: - Angle $14^\circ$ at $C$ between chords $CF$ and $CG$. - Angle $y$ at $G$ between $CG$ and $GB$. - Angle $34^\circ$ outside near $E$ on extension of $CG$. 10. The external angle $34^\circ$ at $E$ equals the sum of the opposite interior angles $14^\circ + y$ (Exterior Angle Theorem): $$34^\circ = 14^\circ + y$$ 11. Solve for $y$: $$y = 34^\circ - 14^\circ = 20^\circ$$ 12. Angle $x$ is not explicitly marked in this part, so no $x$ to solve here. --- **Final answers:** - Part a: $x = 23^\circ$, $y = 23^\circ$ - Part b: $y = 20^\circ$