Two opposite angles of a parallelogram are `(3x-2)^@ and (50-x)^@`. The measures of all its angles are

Options

A. `97^@,83^@,97^@,83^@`
B. `37^@,143^@,37^@,143^@`
C. `76^@,104^@,76^@,104^@`
D. None of these

Answer

To find the measures of all the angles in the given parallelogram, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Opposite Angles**:
In a parallelogram, opposite angles are equal. We are given two opposite angles:
- Angle A = \(3x - 2\) degrees
- Angle C = \(50 - x\) degrees

2. **Set Up the Equation**:
Since Angle A and Angle C are equal, we can set up the equation:
\[ 3x - 2 = 50 - x \]

3. **Solve for x**:
To solve for \(x\), we first add \(x\) to both sides:
\[ 3x + x - 2 = 50 \]
This simplifies to:
\[ 4x - 2 = 50 \]
Next, we add 2 to both sides:
\[ 4x = 52 \]
Now, divide both sides by 4:
\[ x = \frac{52}{4} = 13 \]

4. **Find the Angles**:
Now that we have \(x\), we can find the measures of Angle A and Angle C:
- For Angle A:
\[ \text{Angle A} = 3x - 2 = 3(13) - 2 = 39 - 2 = 37 \text{ degrees} \]
- For Angle C:
\[ \text{Angle C} = 50 - x = 50 - 13 = 37 \text{ degrees} \]

5. **Find the Remaining Angles**:
Since opposite angles are equal, Angle B and Angle D will also be equal. We can find their measures using the property that the sum of adjacent angles in a parallelogram is 180 degrees:
- Angle D + Angle A = 180 degrees
\[ \text{Angle D} + 37 = 180 \]
Subtracting 37 from both sides gives:
\[ \text{Angle D} = 180 - 37 = 143 \text{ degrees} \]
- Since Angle B is opposite to Angle D:
\[ \text{Angle B} = 143 \text{ degrees} \]

6. **Final Angles**:
Now we can summarize the measures of all angles in the parallelogram:
- Angle A = 37 degrees
- Angle B = 143 degrees
- Angle C = 37 degrees
- Angle D = 143 degrees

### Summary of Angles:
- Angle A = 37°
- Angle B = 143°
- Angle C = 37°
- Angle D = 143°