Two adjacent angles of a parallelograms are in the ratio 4:5. Find the measure of each of its angles.

Options

A. `70^@,110^@,70^@,110^@`
B. `60^@,120^@,60^@,120^@`
C. `80^@,100^@,80^@,100^@`
D. `50^@,130^@,50^@,130^@`

Answer

To solve the problem of finding the measures of the angles in a parallelogram where two adjacent angles are in the ratio 4:5, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Angles**:
Let the two adjacent angles of the parallelogram be \( \angle A \) and \( \angle B \). According to the problem, the ratio of these angles is given as \( 4:5 \).

2. **Assigning Variables**:
We can express these angles in terms of a common variable \( x \):
- \( \angle A = 4x \)
- \( \angle B = 5x \)

3. **Using the Property of Parallelograms**:
In a parallelogram, the sum of adjacent angles is always \( 180^\circ \). Therefore, we can write the equation:
\[ \angle A + \angle B = 180^\circ \]

4. **Substituting the Angles**:
Substitute the expressions for \( \angle A \) and \( \angle B \) into the equation:
\[ 4x + 5x = 180^\circ \]

5. **Combining Like Terms**:
Combine the terms on the left side:
\[ 9x = 180^\circ \]

6. **Solving for \( x \)**:
Divide both sides by 9 to find \( x \):
\[ x = \frac{180^\circ}{9} = 20^\circ \]

7. **Finding the Angles**:
Now that we have the value of \( x \), we can find the measures of the angles:
- \( \angle A = 4x = 4 \times 20^\circ = 80^\circ \)
- \( \angle B = 5x = 5 \times 20^\circ = 100^\circ \)

8. **Finding the Remaining Angles**:
In a parallelogram, opposite angles are equal. Therefore:
- \( \angle C = \angle A = 80^\circ \)
- \( \angle D = \angle B = 100^\circ \)

### Final Answer:
The measures of the angles in the parallelogram are:
- \( \angle A = 80^\circ \)
- \( \angle B = 100^\circ \)
- \( \angle C = 80^\circ \)
- \( \angle D = 100^\circ \)