1. **State the problem:** We are given a parallelogram ABCD with angles expressed in terms of $x$: - $m\angle DAB = 85 - 2x$ - $m\angle ABC = 90 + x$ - $m\angle BCD = 80 - 3x$ - $m\angle CDA = 105 + 4x$ We need to find the value of $x$. 2. **Recall properties of parallelograms:** Opposite angles in a parallelogram are equal, and consecutive angles are supplementary (sum to 180 degrees). 3. **Set up equations using opposite angles:** Since $\angle DAB$ and $\angle BCD$ are opposite angles, they are equal: $$85 - 2x = 80 - 3x$$ 4. **Solve for $x$ from opposite angles:** $$85 - 2x = 80 - 3x$$ $$85 - 2x + 3x = 80$$ $$85 + x = 80$$ $$x = 80 - 85$$ $$x = -5$$ 5. **Verify with consecutive angles:** Consecutive angles $\angle DAB$ and $\angle ABC$ are supplementary: $$m\angle DAB + m\angle ABC = 180$$ Substitute expressions: $$ (85 - 2x) + (90 + x) = 180 $$ Simplify: $$ 85 - 2x + 90 + x = 180 $$ $$ 175 - x = 180 $$ Solve for $x$: $$ -x = 180 - 175 $$ $$ -x = 5 $$ $$ x = -5 $$ This matches the previous value, confirming $x = -5$. 6. **Final answer:** $$\boxed{-5}$$