1. **State the problem:** We need to find the values of angles $x$, $y$, and $z$ in a parallelogram with given angles and $x=75^\circ$. 2. **Recall properties of parallelograms:** Opposite angles are equal, and adjacent angles are supplementary (sum to $180^\circ$). Also, diagonals intersect and create angles with special relationships. 3. **Given:** $x=75^\circ$, angles labeled $121^\circ$, $31^\circ$, $74^\circ$, $y^\circ$, and $z^\circ$. 4. **Find $y$:** Since $x$ and $y$ are adjacent angles formed by the diagonals inside the parallelogram, and the sum of angles around a point is $360^\circ$, use the given angles to find $y$. 5. **Calculate $y$:** The angles around the intersection point are $x$, $y$, $74^\circ$, and $31^\circ$. $$x + y + 74 + 31 = 360$$ Substitute $x=75$: $$75 + y + 74 + 31 = 360$$ $$y + 180 = 360$$ $$y = 360 - 180 = 180$$ This is impossible for an angle inside the parallelogram, so re-examine the angle relationships. 6. **Use linear pairs:** Angles $x$ and $121^\circ$ are supplementary because they are adjacent angles on a straight line. $$x + 121 = 180$$ Check with $x=75$: $$75 + 121 = 196 \neq 180$$ So $x$ cannot be $75$ if $121^\circ$ is adjacent. 7. **Since $x=75$ is given, $121^\circ$ must be opposite or non-adjacent. Use the property that opposite angles in parallelogram are equal. So $121^\circ$ is opposite to $z^\circ$. Therefore: $$z = 121$$ 8. **Find $y$ using the triangle formed by angles $y$, $31^\circ$, and $74^\circ$ inside the parallelogram:** Sum of angles in a triangle is $180^\circ$. $$y + 31 + 74 = 180$$ $$y + 105 = 180$$ $$y = 180 - 105 = 75$$ 9. **Summary:** $$x = 75^\circ$$ $$y = 75^\circ$$ $$z = 121^\circ$$