1. **Problem statement:** We have a circle with center O and points D, E, F on the circumference. Given angles are $\angle EDF = x^\circ$ and $\angle OEF = 4x^\circ$. We need to find: (a)(i) $\angle OFE$ in terms of $x$, (a)(ii) $\angle EOF$ in terms of $x$, (b) the value of $x$. 2. **Key facts and formulas:** - The center O and points on the circumference form isosceles triangles because radii are equal. - The sum of angles in any triangle is $180^\circ$. - The angle at the center is twice the angle at the circumference subtending the same arc. 3. **Step (a)(i): Find $\angle OFE$** - Triangle $OEF$ has sides $OE = OF$ (radii), so $\triangle OEF$ is isosceles with $OE = OF$. - Given $\angle OEF = 4x^\circ$, the base angles are $\angle OFE$ and $\angle EOF$. - Let $\angle OFE = y$. - Sum of angles in $\triangle OEF$: $$4x + y + y = 180$$ $$4x + 2y = 180$$ $$2y = 180 - 4x$$ $$y = 90 - 2x$$ - So, $\angle OFE = 90 - 2x^\circ$. 4. **Step (a)(ii): Find $\angle EOF$** - From above, $\angle EOF = y = 90 - 2x^\circ$. 5. **Step (b): Calculate $x$** - Given $\angle EDF = x^\circ$ is an angle at the circumference subtending arc $OF$. - The central angle $\angle EOF$ subtending the same arc is $2x^\circ$ (angle at center is twice angle at circumference). - But from (a)(ii), $\angle EOF = 90 - 2x$. - Equate: $$90 - 2x = 2x$$ $$90 = 4x$$ $$x = 22.5$$ **Final answers:** - $\angle OFE = 90 - 2x = 90 - 2(22.5) = 45^\circ$ - $\angle EOF = 45^\circ$ - $x = 22.5^\circ$