Apply the Tangent-Chord Theorem.

The angle between the tangent $EG$ and the chord $AE$ ($\hat{E}_1$) is equal to the angle subtended by the chord $AE$ in the alternate segment. This angle is $\angle ABE$.

We are given $\angle ABC = 35^\circ$. Since $B$ is a point on the circumference, $\angle ABE$ is the same as $\angle ABC$.