1. **Problem statement:** We have a circle with points A, B, C, D on the circumference. EF is a tangent to the circle at A. AB is parallel to DC. Given angles: \(\angle EAD = 35^\circ\) (between tangent EF and chord AD), and \(\angle BAD = 60^\circ\). We need to find: (a) \(\angle DCB\) with a geometrical reason. (b) \(\angle DBC\). 2. **Recall important rules:** - The angle between a tangent and a chord is equal to the angle in the alternate segment of the circle. - Opposite angles between parallel lines are equal. - Angles subtended by the same chord in the circle are equal. 3. **Find \(\angle DCB\):** - By the alternate segment theorem, \(\angle EAD = \angle DCB = 35^\circ\). 4. **Find \(\angle DBC\):** - Since AB is parallel to DC, \(\angle BAD = \angle DCB = 60^\circ\) (corresponding angles). - But \(\angle BAD = 60^\circ\) is given. - Triangle BCD has angles \(\angle DCB = 35^\circ\) and \(\angle DBC = x\), and \(\angle BDC = y\). - Since AB is parallel to DC, \(\angle DBC = \angle ABC = 60^\circ\) (alternate interior angles). **Final answers:** - \(\angle DCB = 35^\circ\) because the angle between tangent EF and chord AD equals the angle in the alternate segment. - \(\angle DBC = 60^\circ\) because AB is parallel to DC, so alternate interior angles are equal.