If one angle of a parallelogram is `24^@` less than twice the smallest angle then the largest angle of the parallelogram is

Options

A. `68^@`
B. `102^@`
C. `112^@`
D. `176^@`

Answer

To solve the problem step by step, we will follow the logic presented in the video transcript.

### Step-by-Step Solution:

1. **Understanding the Angles of a Parallelogram**:
In a parallelogram, opposite angles are equal, and the sum of all angles is 360 degrees.

2. **Let the Smallest Angle be x**:
We denote the smallest angle of the parallelogram as \( x \).

3. **Express the Other Angle**:
According to the problem, one angle is 24 degrees less than twice the smallest angle. Therefore, we can express this angle as:
\[ 2x - 24 \]

4. **Identify the Angles**:
In a parallelogram, we have two pairs of opposite angles. Thus, we can denote the angles as:
- Angle A = \( x \)
- Angle B = \( 2x - 24 \)
- Angle C = \( x \) (opposite to A)
- Angle D = \( 2x - 24 \) (opposite to B)

5. **Set Up the Equation**:
The sum of all angles in a parallelogram is 360 degrees. Therefore, we can write:
\[ x + (2x - 24) + x + (2x - 24) = 360 \]

6. **Simplify the Equation**:
Combine like terms:
\[ x + x + 2x + 2x - 24 - 24 = 360 \]
This simplifies to:
\[ 6x - 48 = 360 \]

7. **Solve for x**:
Add 48 to both sides:
\[ 6x = 360 + 48 \]
\[ 6x = 408 \]
Now divide by 6:
\[ x = \frac{408}{6} = 68 \]

8. **Find the Largest Angle**:
Now that we have \( x \), we can find the other angle:
\[ 2x - 24 = 2(68) - 24 = 136 - 24 = 112 \]
Thus, the largest angle of the parallelogram is \( 112^\circ \).

### Conclusion:
The largest angle of the parallelogram is **112 degrees**.