1. **Problem Statement:** Given a circle with center O and points A, B, C, D, E on the circumference, line FAG is tangent at A, chord DE extended meets FAG at F. Given angles: $\angle FAE=30^\circ$, $\angle EDC=110^\circ$, and $\angle OCB=55^\circ$. Find the sizes of requested angles. 2. **Key Theorems and Rules:** - Tangent-Chord Angle Theorem: The angle between a tangent and a chord through the point of contact equals the angle in the alternate segment. - Opposite angles in cyclic quadrilaterals sum to $180^\circ$. - Angles subtended by the same chord at the circumference are equal. - Central angle is twice the inscribed angle subtending the same arc. 3. **Step-by-step Solution:** - From Tangent-Chord Theorem, $\angle FAE = 30^\circ$ equals the angle in the alternate segment, so $\angle ADE = 30^\circ$. - Given $\angle EDC = 110^\circ$, and $\angle ADE = 30^\circ$, angles at D on chord DE sum to $\angle EDC + \angle ADE = 110^\circ + 30^\circ = 140^\circ$. - Since points lie on the circle, $\angle EDC$ and $\angle EAC$ subtend the same arc EC, so $\angle EAC = 110^\circ$. - Given $\angle OCB = 55^\circ$, which is a central angle, the inscribed angle $\angle OAB$ subtending the same arc is half, so $\angle OAB = 27.5^\circ$. - Using these, other angles can be found by applying cyclic quadrilateral properties and angle sums. 4. **Final answers:** - $\angle ADE = 30^\circ$ - $\angle EAC = 110^\circ$ - $\angle OAB = 27.5^\circ$ These are the key angles derived from the given information and circle theorems.