1. **State the problem:** We are given a circle with center O and points A, B, C on the circumference. The tangent DB is extended to T, with angle TBA = 62°. We need to find the values of $x$ and $y$ (assumed to be angles related to the diagram). 2. **Recall key properties:** - The angle between a tangent and a chord through the point of contact equals the angle in the alternate segment. - The angle at the center is twice the angle at the circumference subtended by the same chord. 3. **Find $x$:** - Given angle TBA = 62°, and DB is tangent at B, angle DBA (between tangent and chord BA) = 62°. - By alternate segment theorem, angle BCA (angle $x$) = 62°. 4. **Find $y$:** - Angle at center O subtended by chord AC is $y$. - Angle at circumference subtended by chord AC is angle $\gamma$. - By circle theorem, $y = 2\gamma$. - Given angle at D is 40°, and since DB is tangent, angle between tangent and chord BC equals angle BCA (which is $x=62°$), so $\gamma = 40°$. - Therefore, $y = 2 \times 40° = 80°$. **Final answers:** $$x = 62°$$ $$y = 80°$$