If an angle of a parallelogram is two-third of its adjacent angle, find the smallest angle of the parallelogram.

Options

A. ` 108 ^0 `
B. ` 72 ^0 `
C. ` 90 ^0 `
D. none of these

Answer

To solve the problem step by step, we will follow the reasoning provided in the video transcript.

### Step-by-Step Solution:

1. **Understanding the Problem:**
We are given that one angle of a parallelogram is two-thirds of its adjacent angle. Let's denote the angles of the parallelogram as \( A \) and \( B \).

2. **Setting Up the Equation:**
According to the properties of a parallelogram, the sum of two adjacent angles is 180 degrees. Therefore, we can write:
\[ A + B = 180^\circ \]
Also, it is given that:
\[ A = \frac{2}{3}B \]

3. **Substituting the Value of A:**
We can substitute the expression for \( A \) into the equation for the sum of the angles:
\[ \frac{2}{3}B + B = 180^\circ \]

4. **Finding a Common Denominator:**
To combine the terms on the left side, we can express \( B \) as \( \frac{3}{3}B \):
\[ \frac{2}{3}B + \frac{3}{3}B = 180^\circ \]
This simplifies to:
\[ \frac{5}{3}B = 180^\circ \]

5. **Solving for B:**
To isolate \( B \), we multiply both sides by 3:
\[ 5B = 540^\circ \]
Now, divide by 5:
\[ B = \frac{540^\circ}{5} = 108^\circ \]

6. **Finding A:**
Now that we have \( B \), we can find \( A \) using the relationship \( A = \frac{2}{3}B \):
\[ A = \frac{2}{3} \times 108^\circ = \frac{216^\circ}{3} = 72^\circ \]

7. **Conclusion:**
The angles of the parallelogram are:
- \( A = 72^\circ \)
- \( B = 108^\circ \)

Since opposite angles in a parallelogram are equal:
- Angle \( C = A = 72^\circ \)
- Angle \( D = B = 108^\circ \)

Therefore, the smallest angle of the parallelogram is \( 72^\circ \).