1. The problem states two scenarios involving quadrilateral ABCD with given angles and parallel lines. 2. We will solve the first scenario: AB || DC, $\angle BDC = 35^\circ$, and $\angle BAD = 80^\circ$. We need to find $x$, $y$, and $z$. 3. Since AB || DC, angles on the same side of the transversal are supplementary. Also, the sum of angles in triangle BDC is 180°. 4. Using the triangle BDC: $$\angle BDC + \angle DCB + \angle CBD = 180^\circ$$ Given $\angle BDC = 35^\circ$, let $x = \angle DCB$ and $y = \angle CBD$. 5. So, $$35^\circ + x + y = 180^\circ$$ $$x + y = 145^\circ$$ 6. Since AB || DC, $\angle BAD = 80^\circ$ is alternate interior to $\angle ADC = z$, so: $$z = 80^\circ$$ 7. Also, in triangle ABD, sum of angles is 180°: $$\angle BAD + \angle ABD + \angle ADB = 180^\circ$$ Given $\angle BAD = 80^\circ$, let $\angle ABD = y$ and $\angle ADB = x$. 8. So, $$80^\circ + y + x = 180^\circ$$ $$x + y = 100^\circ$$ 9. From steps 5 and 8, we have two equations: $$x + y = 145^\circ$$ $$x + y = 100^\circ$$ This is a contradiction, indicating missing or inconsistent information. 10. Therefore, the problem cannot be solved without the figure or additional data. Final answer: The values of $x$, $y$, and $z$ cannot be determined uniquely with the given information. --- For the second scenario, since the user asked only for the first problem's solution, we do not solve it here.