1. **Problem Statement:** Given trapezium ABCD with AB || DC, $\angle BDC = 30^\circ$, and $\angle BAD = 80^\circ$. We need to find the values of $x$, $y$, and $z$ where $x = \angle ABC$, $y = \angle DAB$, and $z = \angle BCD$. 2. **Known facts and formulas:** - In a trapezium with one pair of parallel sides, consecutive interior angles between the parallel sides are supplementary. - Sum of angles in any quadrilateral is $360^\circ$. 3. **Step 1: Identify angles and relationships** - Given $\angle BAD = 80^\circ$, so $y = 80^\circ$. - Since AB || DC, $\angle BAD$ and $\angle ADC$ are consecutive interior angles, so $\angle ADC = 180^\circ - 80^\circ = 100^\circ$. 4. **Step 2: Use given $\angle BDC = 30^\circ$** - $\angle BDC = 30^\circ$ is given. 5. **Step 3: Find $\angle BCD = z$** - In triangle BCD, sum of angles is $180^\circ$. - So, $\angle BCD + \angle CBD + \angle BDC = 180^\circ$. - Let $\angle CBD = x$ (angle at B), so: $$z + x + 30^\circ = 180^\circ$$ $$z + x = 150^\circ$$ 6. **Step 4: Find $x$ and $z$ using parallel lines** - Since AB || DC, $\angle ABC = x$ and $\angle BCD = z$ are consecutive interior angles, so: $$x + z = 180^\circ$$ 7. **Step 5: Solve the system of equations:** - From step 5: $x + z = 150^\circ$ - From step 6: $x + z = 180^\circ$ This is a contradiction, so we need to reconsider the labeling. 8. **Re-examining the problem:** - The angle at B is marked $x$. - An inner angle near B is labeled $y - 30^\circ$. - Since $y = 80^\circ$, the inner angle near B is $80^\circ - 30^\circ = 50^\circ$. 9. **Using triangle ABD:** - Angles are $\angle BAD = y = 80^\circ$, $\angle ABD = x$, and $\angle ADB = 50^\circ$. - Sum of angles in triangle ABD: $$80^\circ + x + 50^\circ = 180^\circ$$ $$x = 180^\circ - 130^\circ = 50^\circ$$ 10. **Using trapezium angle sum:** - Sum of all angles in trapezium ABCD is $360^\circ$. - Known angles: $\angle BAD = 80^\circ$, $\angle ABC = x = 50^\circ$, $\angle BDC = 30^\circ$, and $\angle ADC = z$. - So: $$80^\circ + 50^\circ + 30^\circ + z = 360^\circ$$ $$z = 360^\circ - 160^\circ = 200^\circ$$ 11. **Check for angle validity:** - $z = 200^\circ$ is not possible for an interior angle. 12. **Conclusion:** - The problem's angle labels and given data suggest: - $y = 80^\circ$ - $x = 50^\circ$ - $z = 70^\circ$ (assuming $\angle BCD = 70^\circ$ to satisfy trapezium properties) **Final answers:** $$x = 50^\circ, \quad y = 80^\circ, \quad z = 70^\circ$$