If an angle of a parallelogram is two-thirds of its adjacent angle, the smallest angle of the parallelogram is

Options

A. `54^@`
B. `72^@`
C. `81^@`
D. `108^@`

Answer

To solve the problem, we need to find the smallest angle of a parallelogram where one angle is two-thirds of its adjacent angle. Let's denote the angles of the parallelogram as follows:

- Let angle A = x (the angle we are considering)
- Then, angle B (the adjacent angle) = (2/3)x

Since opposite angles in a parallelogram are equal, we have:

- Angle C = x
- Angle D = (2/3)x

Now, we can express the sum of the angles in the parallelogram:

1. **Step 1: Write the equation for the sum of angles.**
The sum of all angles in a parallelogram is 360 degrees. Therefore, we can write:
\[ A + B + C + D = 360^\circ \]
Substituting the values we have:
\[ x + \frac{2}{3}x + x + \frac{2}{3}x = 360^\circ \]

2. **Step 2: Combine like terms.**
Combine the terms on the left side:
\[ 2x + \frac{4}{3}x = 360^\circ \]
To combine these, we need a common denominator. The common denominator for 3 is 3:
\[ \frac{6}{3}x + \frac{4}{3}x = 360^\circ \]
This simplifies to:
\[ \frac{10}{3}x = 360^\circ \]

3. **Step 3: Solve for x.**
To find x, multiply both sides by 3:
\[ 10x = 360 \times 3 \]
\[ 10x = 1080 \]
Now, divide both sides by 10:
\[ x = \frac{1080}{10} = 108^\circ \]

4. **Step 4: Find the smallest angle.**
The smallest angle of the parallelogram is:
\[ \text{Smallest angle} = \frac{2}{3}x = \frac{2}{3} \times 108^\circ \]
Calculating this gives:
\[ = \frac{216}{3} = 72^\circ \]

Thus, the smallest angle of the parallelogram is **72 degrees**.