Apply the Tangent-Chord Theorem.
This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
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3.1 Calculate $\hat{E}_1$
Step 1: 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 $\angle ABE$.
$$ \hat{E}_1 = \angle ABE \quad (\text{Tangent-Chord Theorem}) $$
Step 2: Use the given angle.
We are given $\angle ABC = 35^\circ$. Since $B$ is a point on the circumference, $\angle ABE$ is the same as $\angle ABC$.
