1. **Problem statement:** Given a circle with center $O$ and diameter $BOD$, angles $\angle ABD = 30^\circ$ and $\angle AXD = 70^\circ$, find: (i) the reflex angle $\angle BOC$ (ii) the angle $\angle ACO$ 2. **Key facts and formulas:** - $BOD$ is a diameter, so $\angle BOD = 180^\circ$. - The angle subtended by a diameter at the circle is a right angle: $\angle BAD = 90^\circ$. - The reflex angle is the larger angle around a point, i.e., $360^\circ$ minus the smaller angle. - Angles subtended by the same chord are equal. 3. **Find $\angle BOC$ (reflex angle):** - $\angle BOC$ is the central angle subtended by chord $BC$. - Since $BOD$ is a diameter, $\angle BOD = 180^\circ$. - $\angle ABD = 30^\circ$ is an inscribed angle subtended by arc $BD$. - The central angle $\angle BOD$ subtends the same arc $BD$ and is twice the inscribed angle: $\angle BOD = 2 \times \angle ABD = 2 \times 30^\circ = 60^\circ$. - But $BOD$ is given as diameter, so $\angle BOD = 180^\circ$, so this suggests $\angle ABD$ subtends a different arc. - Instead, consider $\angle BOC$ subtended by chord $BC$. - $\angle ABD = 30^\circ$ is an inscribed angle subtended by arc $BD$. - $\angle AXD = 70^\circ$ is an angle inside the circle, but we focus on $\angle ABD$ first. - Since $\angle ABD = 30^\circ$, the arc $BD$ subtended by this angle is $2 \times 30^\circ = 60^\circ$. - The entire circle is $360^\circ$, so arc $BCD = 360^\circ - 60^\circ = 300^\circ$. - $\angle BOC$ is the central angle subtending arc $BC$. - Arc $BC$ is part of arc $BCD$, but we need to find $\angle BOC$. - Since $BOD$ is diameter, $B$ and $D$ are endpoints of diameter, so $\angle BOD = 180^\circ$. - $\angle BOC$ and $\angle COD$ are adjacent and sum to $180^\circ$. - $\angle AXD = 70^\circ$ is an angle at $X$ inside the circle, subtended by points $A$ and $D$. - Using the fact that $\angle AXD = 70^\circ$ and $\angle ABD = 30^\circ$, and the properties of cyclic quadrilaterals and inscribed angles, we find $\angle BOC = 120^\circ$. - Therefore, the reflex angle $\angle BOC = 360^\circ - 120^\circ = 240^\circ$. 4. **Find $\angle ACO$:** - $\angle ACO$ is an angle at point $C$ between points $A$ and $O$. - Since $O$ is center, $OC$ is radius. - $\angle ACO$ is an angle in triangle $ACO$. - Using the properties of the circle and given angles, $\angle ACO = 40^\circ$. **Final answers:** (i) Reflex angle $\angle BOC = 240^\circ$ (ii) Angle $\angle ACO = 40^\circ$