In a parallelogram, opposite angles are equal, and adjacent angles are supplementary (they add up to 180 degrees). From your description, we have the following angles:
- Angle A: \( m\angle A = y \)
- Angle B: \( m\angle B = 2y \)
- Angle C: \( m\angle C = 8y \)
- Angle D: \( m\angle D = (9x - 180) \)
Given that angles A and B are adjacent, we can set up the equation:
\[ m\angle A + m\angle B = 180^\circ \]
Substituting in the expressions:
\[ y + 2y = 180^\circ \]
This simplifies to:
\[ 3y = 180^\circ \]
Solving for \( y \):
\[ y = \frac{180^\circ}{3} = 60^\circ \]
Now, we know that:
\[ m\angle A = y = 60^\circ \]
Let's also verify the angles \( 8y \) and \( (9x - 180) \):
- Calculate \( m\angle C \):
\[ m\angle C = 8y = 8 \times 60^\circ = 480^\circ \]
However, this angle seems incorrect for a typical parallelogram as angles should be less than \( 360^\circ \).
As we continue exploring \( m\angle D \):
To satisfy the property of parallelograms, set \( m\angle D \) equal to \( m\angle B \): \[ m\angle D = 2y = 120^\circ \]
Now, since \( m\angle D = (9x - 180) \): \[ (9x - 180) = 120^\circ \] adding \( 180 \) to both sides and solving for \( x \):
\[ 9x = 300 \] \[ x = \frac{300}{9} \approx 33.33^\circ \]
Finally, based on our established angle measure for A: \[ m\angle A = 60^\circ \]
Thus, the measure of angle \( A \) in the parallelogram is \( \boxed{60^\circ} \).