If an angle of a parallelogram is two-third of its adjacent angle, then smallest angle of the parallelogram is :

Question

Accepted Answer

To solve the problem of finding the smallest angle of a parallelogram where one angle is two-thirds of its adjacent angle, we need to use the properties of a parallelogram and some basic algebra.

### Explanation:

1. **Properties of a Parallelogram:**
   - Opposite angles in a parallelogram are equal.
   - The sum of adjacent angles in a parallelogram is 180°.

2. **Setting Up the Equation:**
   - Let one angle of the parallelogram be $ x $.
   - According to the problem, the adjacent angle is two-thirds of $ x $, so it can be expressed as $ \frac{2}{3}x $.

3. **Using the Sum of Adjacent Angles:**
   - The sum of these two adjacent angles is 180°.
   - Therefore, we can set up the equation: 
     $$
     x + \frac{2}{3}x = 180°
     $$

4. **Solving the Equation:**
   - Combine the terms: 
     $$
     \frac{3}{3}x + \frac{2}{3}x = 180°
     $$
     $$
     \frac{5}{3}x = 180°
     $$
   - Multiply both sides by 3 to clear the fraction:
     $$
     5x = 540°
     $$
   - Divide by 5 to solve for $ x $:
     $$
     x = 108°
     $$

5. **Finding the Smaller Angle:**
   - The adjacent angle, which is two-thirds of $ x $, is:
     $$
     \frac{2}{3} \times 108° = 72°
     $$

6. **Conclusion:**
   - The smallest angle of the parallelogram is 72°, which corresponds to Option 4.

### Why Other Options are Incorrect:

- **Option 1 (140°):** This cannot be the smallest angle because it is larger than 108°, which is the calculated value of $ x $.
- **Option 2 (108°):** This is the value of $ x $, the larger angle, not the smallest angle.
- **Option 3 (54°):** This value is incorrect because it does not satisfy the condition that one angle is two-thirds of its adjacent angle.

Thus, the correct answer is Option 4, 72°, as it is the smallest angle in the given conditions of the parallelogram.