1. **Problem statement:** Given a circle with center O, diameter BOD, chord TC intersecting BOD at Z, and tangent XTY at T. Given angles: $\angle TAB = 124^\circ$ and $\angle CBD = 28^\circ$. Find angles: (a) $\angle CTD$, (b) $\angle BDT$, (c) $\angle BOT$, (d) $\angle DBT$ with reasons. 2. **Key facts and formulas:** - Angle between tangent and chord equals the angle in the alternate segment. - Angles subtended by the same chord in the circle are equal. - Diameter subtends a right angle to any point on the circle. - Opposite angles in cyclic quadrilaterals sum to $180^\circ$. 3. **Step (a) Find $\angle CTD$:** - By the tangent-chord theorem, $\angle CTD = \angle CAB$ (angle in alternate segment). - Given $\angle TAB = 124^\circ$, and $\angle TAB = \angle CAB + \angle BAT$. - Since $\angle BAT$ is on the circle and $\angle TAB$ is given, we consider $\angle CTD = 124^\circ$ by alternate segment theorem. 4. **Step (b) Find $\angle BDT$:** - $\angle BDT$ and $\angle CBD$ subtend the same chord BD. - Given $\angle CBD = 28^\circ$, so $\angle BDT = 28^\circ$ (angles subtended by same chord). 5. **Step (c) Find $\angle BOT$:** - $\angle BOT$ is the central angle subtending arc BT. - $\angle BDT$ is an inscribed angle subtending the same arc BT. - Central angle is twice the inscribed angle: $\angle BOT = 2 \times \angle BDT = 2 \times 28^\circ = 56^\circ$. 6. **Step (d) Find $\angle DBT$:** - In triangle BDT, sum of angles is $180^\circ$. - Known angles: $\angle BDT = 28^\circ$, $\angle CTD = 124^\circ$ (from step a, but CTD is not in triangle BDT, so ignore). - Use cyclic quadrilateral properties or supplementary angles. - Since $\angle TAB = 124^\circ$ and $\angle DBT$ is adjacent, $\angle DBT = 180^\circ - 124^\circ = 56^\circ$. **Final answers:** (a) $\angle CTD = 124^\circ$ (b) $\angle BDT = 28^\circ$ (c) $\angle BOT = 56^\circ$ (d) $\angle DBT = 56^\circ$