1. **State the problem:** We have a circle with center O and points A, B, C, D on the circumference. Given: - AOD is a diameter. - OB is parallel to DC. - Angle BÔD = 140°. Find the angles $t$, $y$, $x$, and $z$. 2. **Recall important facts:** - The diameter subtends a right angle to any point on the circle, so any angle subtended by AOD on the circumference is 90°. - Parallel lines imply equal alternate interior angles. - The sum of angles around point O is 360°. 3. **Analyze angle BÔD = 140°:** Since $\angle BÔD = 140^\circ$, the remaining angle around O on the diameter line AOD is $360^\circ - 140^\circ = 220^\circ$. 4. **Find angle AÔB:** Since AOD is a straight line (diameter), $\angle AÔD = 180^\circ$. Given $\angle BÔD = 140^\circ$, then $\angle AÔB = 180^\circ - 140^\circ = 40^\circ$. 5. **Find angle $t$ at B:** Angle $t$ is the angle at B subtended by points A and D on the circle. By the circle theorem, the angle at the circumference subtended by the same chord is half the angle at the center. So, $t = \frac{1}{2} \times \angle AÔD = \frac{1}{2} \times 180^\circ = 90^\circ$. 6. **Find angle $y$ at C:** Since OB is parallel to DC, angle $y$ at C equals angle $t$ at B (alternate interior angles). Therefore, $y = t = 90^\circ$. 7. **Find angles $x$ and $z$ at D:** Angles $x$ and $z$ at D are angles on the circle subtended by chords BC and BD. Since AOD is diameter, angle $x$ (between C, D, B) is $90^\circ$ (angle in a semicircle). Angle $z$ is supplementary to angle $x$ because they form a straight line at D. So, $z = 180^\circ - x = 180^\circ - 90^\circ = 90^\circ$. **Final answers:** $$t = 90^\circ, \quad y = 90^\circ, \quad x = 90^\circ, \quad z = 90^\circ.$$