1. **State the problem:** We have a parallelogram ABCD with sides labeled as follows: AB = 12x, BC = 10y + 10, CD is parallel to AB, and DA = 6x. We need to find the value of $y$. 2. **Recall the property of parallelograms:** Opposite sides of a parallelogram are equal in length. Therefore, AB = CD and BC = DA. 3. **Set up the equation for the sides BC and DA:** Since BC and DA are opposite sides, they must be equal: $$10y + 10 = 6x$$ 4. **Set up the equation for the sides AB and CD:** Since AB and CD are opposite sides, they must be equal: $$12x = CD$$ But CD is not given explicitly, so we focus on the equation involving $y$ and $x$. 5. **Use the fact that AB and CD are equal and parallel, and DA and BC are equal and parallel. Since AB = 12x and DA = 6x, the parallelogram's sides suggest $x$ is consistent. We can solve for $x$ from the equality of AB and CD, but since CD is not given, we use the equality of BC and DA to find $y$ in terms of $x$. 6. **Assuming the parallelogram is valid, the problem likely expects $x$ to be such that the sides match. Since the problem only asks for $y$, we can express $y$ in terms of $x$ from step 3:** $$10y + 10 = 6x$$ 7. **Solve for $y$:** $$10y = 6x - 10$$ $$y = \frac{6x - 10}{10} = \frac{6x}{10} - 1 = 0.6x - 1$$ 8. **Use the fact that AB = 12x and DA = 6x, so the parallelogram's sides are consistent. Since AB and CD are equal, and CD is parallel to AB, the problem likely implies $x$ is such that the sides are consistent. To find a numeric value for $y$, we need a numeric value for $x$. Since the problem provides multiple choice answers for $y$, we can test these values by substituting back into the equation $10y + 10 = 6x$ and checking if $x$ is consistent with the other side lengths. 9. **Test each option for $y$:** - For $y=10$: $$10(10) + 10 = 100 + 10 = 110 = 6x \Rightarrow x = \frac{110}{6} \approx 18.33$$ - For $y=11$: $$10(11) + 10 = 110 + 10 = 120 = 6x \Rightarrow x = \frac{120}{6} = 20$$ - For $y=60$: $$10(60) + 10 = 600 + 10 = 610 = 6x \Rightarrow x = \frac{610}{6} \approx 101.67$$ - For $y=120$: $$10(120) + 10 = 1200 + 10 = 1210 = 6x \Rightarrow x = \frac{1210}{6} \approx 201.67$$ 10. **Check if $x$ values correspond to the other side length AB = 12x. Since AB = 12x, for $x=20$, AB = 240, and DA = 6x = 120. The sides AB and DA are 240 and 120 respectively, which is consistent with the parallelogram's side labels 12x and 6x. 11. **Therefore, the value of $y$ that makes the parallelogram consistent is $y=11$.**