1. **Problem Statement:** We are given a circle with center $O$ and radius 2. Points $A, B, C, D, E$ lie on or inside the circle. Line segments $AC$ and $BD$ intersect at $E$. Given angles are:\ - Angle at $C$ subtended by arc $AB$ is $105^\circ$.\ - Angle adjacent to $E$ on segment $AC$ is $40^\circ$.\ - Angle at $B$ subtended by arc $AD$ is $30^\circ$.\ We need to find the angle $y$ corresponding to the arc from $D$ to $A$.\ \ 2. **Key Theorems and Formulas:**\ - The measure of an inscribed angle is half the measure of its intercepted arc.\ - The sum of arcs around the circle is $360^\circ$.\ - Angles subtended by the same chord are equal.\ \ 3. **Step-by-step solution:**\ - The angle at $C$ subtended by arc $AB$ is $105^\circ$, so the arc $AB$ measures $2 \times 105^\circ = 210^\circ$.\ - The angle at $B$ subtended by arc $AD$ is $30^\circ$, so the arc $AD$ measures $2 \times 30^\circ = 60^\circ$.\ - The total circle is $360^\circ$, so the remaining arc $BD$ plus arc $DA$ plus arc $AB$ must sum to $360^\circ$. We know $AB = 210^\circ$ and $AD = 60^\circ$, so arc $BD = 360^\circ - 210^\circ - 60^\circ = 90^\circ$.\ - The angle $y$ corresponds to the arc from $D$ to $A$, which we found to be $60^\circ$.\ \ 4. **Answer:**\ $$y = 60^\circ$$\ \ This means the arc from $D$ to $A$ measures $60^\circ$.