Question

Find the measure of angle∠A in the parallelogram to the right.
Upper BB2 y degrees2y°8 y degrees8y°x degreesx°left parenthesis 9 x minus 180 right parenthesis degrees(9x−180)°
C
D
A
mangle∠Aequals=

Answer 1

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 \( (9x - 180) \):

1. 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 \( \boxed{60^\circ} \).