1. **Problem statement:** Calculate the angles: i. $\angle E\hat{D}B$ ii. $\angle B\hat{E}C$ iii. $\angle C\hat{B}T$ iv. $\angle B\hat{A}E$ Given: - $ABT$ is tangent to circle $BEDC$ at $B$. - $AEC$ is a straight line through center $O$. - $\angle E\hat{C}B = 35^\circ$. 2. **Key rules and formulas:** - The angle between a tangent and a chord is equal to the angle in the alternate segment. - 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. - Straight line angles sum to $180^\circ$. 3. **Find $\angle E\hat{D}B$:** - $E\hat{D}B$ and $E\hat{C}B$ subtend the same chord $EB$. - Angles in the same segment are equal. - Therefore, $\angle E\hat{D}B = \angle E\hat{C}B = 35^\circ$. 4. **Find $\angle B\hat{E}C$:** - $AEC$ is a straight line through $O$. - $\angle E\hat{C}B = 35^\circ$ is given. - $\angle B\hat{E}C$ subtends chord $BC$. - $\angle B\hat{E}C$ and $\angle B\hat{C}E$ are angles in the same segment. - Since $\angle E\hat{C}B = 35^\circ$, $\angle B\hat{E}C = 35^\circ$. - Also, $\angle B\hat{E}C$ and $\angle B\hat{A}E$ are supplementary as $AEC$ is a straight line. 5. **Find $\angle C\hat{B}T$:** - $ABT$ is tangent at $B$. - Angle between tangent and chord $BC$ is equal to angle in alternate segment $E\hat{C}B$. - So, $\angle C\hat{B}T = \angle E\hat{C}B = 35^\circ$. 6. **Find $\angle B\hat{A}E$:** - $AEC$ is a straight line, so $\angle B\hat{A}E + \angle B\hat{E}C = 180^\circ$. - From step 4, $\angle B\hat{E}C = 35^\circ$. - Therefore, $\angle B\hat{A}E = 180^\circ - 35^\circ = 145^\circ$. **Final answers:** $$\angle E\hat{D}B = 35^\circ$$ $$\angle B\hat{E}C = 35^\circ$$ $$\angle C\hat{B}T = 35^\circ$$ $$\angle B\hat{A}E = 145^\circ$$