Question: 2. In the diagram, TAB is a tangent to the circle with centre $O$. Given that angle $CDE = 110^\circ$ and angle $ACD = angle AEC = 65^\circ$. Find (i) angle $CAE$, (ii) angle $AOC$

1. **Problem statement:** We have a circle with center $O$ and a tangent line $TAB$ touching the circle at point $A$. Given angles $CDE = 110^\circ$, $ACD = 65^\circ$, and $AEC = 65^\circ$, we need to find (i) angle $CAE$ and (ii) angle $AOC$. 2. **Key properties and formulas:** - The tangent to a circle is perpendicular to the radius at the point of tangency, so $\angle OAB = 90^\circ$. - The sum of angles in a triangle is $180^\circ$. - Angles subtended by the same chord in the circle are equal. - The angle at the center is twice the angle at the circumference subtended by the same chord. 3. **Find angle $CAE$:** - Since $ACD = 65^\circ$ and $AEC = 65^\circ$, points $C$, $A$, and $E$ lie on the circle with chords $AC$ and $AE$. - Triangle $CAE$ has angles $CAE$, $AEC = 65^\circ$, and $ACE$. - Using the cyclic quadrilateral property and given angles, we find $\angle CAE = 50^\circ$. 4. **Find angle $AOC$:** - Angle $AOC$ is the central angle subtended by chord $AC$. - The angle at the center is twice the angle at the circumference subtended by the same chord. - Since $\angle ACE = 65^\circ$, $\angle AOC = 2 \times 65^\circ = 130^\circ$. 5. **Summary of answers:** - (i) $\angle CAE = 50^\circ$ - (ii) $\angle AOC = 130^\circ$