The measures of two consecutive angles of a parallelogram are in the ratio 5:4. What is the measure of an acute angle of the parallelogram?

To find the measure of an acute angle of the parallelogram, we first need to find the measures of the consecutive angles.

Let's assume the measure of the first angle is 5x, and the measure of the second angle is 4x, where x is a common factor.

In a parallelogram, opposite angles are congruent. Therefore, the sum of the consecutive angles is 180 degrees.

So, we can write the equation 5x + 4x = 180.

Simplifying the equation, we get 9x = 180.

Dividing both sides by 9, we find x = 20.

Now, we can substitute the value of x back into the expression for the measure of the first angle: 5x = 5 * 20 = 100 degrees.

Since an acute angle is less than 90 degrees, the measure of an acute angle in the parallelogram is 100 degrees.

In a parallelogram, opposite angles are congruent, meaning they have the same measure. Since consecutive angles are given in a ratio of 5:4, we can set up the equation:

5x + 4x = 180

Combining like terms:
9x = 180

Dividing both sides by 9:
x = 20

Now, we can find the measures of the angles.
The first angle is 5x = 5(20) = 100 degrees.
The second angle is 4x = 4(20) = 80 degrees.

The acute angle of the parallelogram is the smaller angle, which in this case is 80 degrees.