1. **Problem statement:** In the circle with center O, points A, B, C, D lie on the circumference, and A, O, X, C, E are collinear. Given $\angle BOC = 48^\circ$ and $\angle CAD = 36^\circ$, find: (i) $\angle DBC$ (ii) $\angle DEC$ (iii) $\angle BCD$ --- 2. **Step 1: Understand the problem and given data.** - $O$ is the center of the circle. - $A, B, C, D$ lie on the circumference. - $A, O, X, C, E$ are collinear. - $DE$ is tangent to the circle at $D$. - $\angle BOC = 48^\circ$ (central angle). - $\angle CAD = 36^\circ$. --- 3. **Step 2: Use circle theorems and angle properties.** - The angle at the center $\angle BOC = 48^\circ$ subtends arc $BC$. - The angle at the circumference subtending the same arc $BC$ is half the central angle. - Tangent and radius are perpendicular at the point of contact. --- 4. **Step 3: Find $\angle DBC$.** - $\angle BOC = 48^\circ$ is the central angle subtending arc $BC$. - $\angle BDC$ at the circumference subtending the same arc is half of $48^\circ$: $$\angle BDC = \frac{1}{2} \times 48^\circ = 24^\circ$$ - Since $D, B, C$ are points on the circumference, $\angle DBC = \angle BDC = 24^\circ$ (angles in the same segment are equal). --- 5. **Step 4: Find $\angle DEC$.** - $DE$ is tangent at $D$, so $\angle EDC = 90^\circ$ because radius $OD$ is perpendicular to tangent. - $\angle BOC = 48^\circ$ subtends arc $BC$, so arc $BC = 48^\circ$. - $\angle BDC = 24^\circ$ as above. - Triangle $DCE$ has angles $\angle DEC, \angle EDC, \angle DCE$. - Using the tangent-secant theorem, $\angle DEC = \angle BCD$ (alternate segment theorem). --- 6. **Step 5: Find $\angle BCD$.** - $\angle CAD = 36^\circ$ is given. - Since $A, O, X, C, E$ are collinear, $\angle CAD$ is an angle between chord $AD$ and line $AC$. - Using the alternate segment theorem, $\angle BCD = \angle CAD = 36^\circ$. --- 7. **Step 6: Summarize answers with reasons:** - (i) $\angle DBC = 24^\circ$ (angle in the same segment is half the central angle). - (ii) $\angle DEC = 36^\circ$ (alternate segment theorem). - (iii) $\angle BCD = 36^\circ$ (given $\angle CAD$ and alternate segment theorem). --- **Final answers:** - $\angle DBC = 24^\circ$ - $\angle DEC = 36^\circ$ - $\angle BCD = 36^\circ$