If the adjacent angles of a parallelogram are in the ratio `1/3:1/2.` Find all the angles of parallelogram.

Answer

To solve the problem of finding the angles of a parallelogram where the adjacent angles are in the ratio \( \frac{1}{3} : \frac{1}{2} \), we can follow these steps:

### Step 1: Define the Angles
Let the adjacent angles of the parallelogram be represented as:
- Angle A = \( \frac{1}{3}X \)
- Angle B = \( \frac{1}{2}X \)

Since opposite angles in a parallelogram are equal, we have:
- Angle C = Angle A = \( \frac{1}{3}X \)
- Angle D = Angle B = \( \frac{1}{2}X \)

### Step 2: Set Up the Equation
The sum of all angles in a quadrilateral (including a parallelogram) is \( 360^\circ \). Therefore, we can write the equation:
\[ \text{Angle A} + \text{Angle B} + \text{Angle C} + \text{Angle D} = 360^\circ \]
Substituting the values we defined:
\[ \frac{1}{3}X + \frac{1}{2}X + \frac{1}{3}X + \frac{1}{2}X = 360^\circ \]

### Step 3: Combine Like Terms
Combining the terms on the left side:
\[ \left(\frac{1}{3}X + \frac{1}{3}X\right) + \left(\frac{1}{2}X + \frac{1}{2}X\right) = 360^\circ \]
This simplifies to:
\[ \frac{2}{3}X + X = 360^\circ \]
To combine the fractions:
\[ \frac{2}{3}X + \frac{3}{3}X = \frac{5}{3}X \]
Thus, we have:
\[ \frac{5}{3}X = 360^\circ \]

### Step 4: Solve for X
To find \( X \), multiply both sides by \( \frac{3}{5} \):
\[ X = 360^\circ \times \frac{3}{5} = 216^\circ \]

### Step 5: Calculate the Angles
Now we can find the measures of the angles:
- Angle A = \( \frac{1}{3}X = \frac{1}{3} \times 216^\circ = 72^\circ \)
- Angle B = \( \frac{1}{2}X = \frac{1}{2} \times 216^\circ = 108^\circ \)

Since Angle C = Angle A and Angle D = Angle B, we have:
- Angle C = \( 72^\circ \)
- Angle D = \( 108^\circ \)

### Final Answer
The angles of the parallelogram are:
- Angle A = \( 72^\circ \)
- Angle B = \( 108^\circ \)
- Angle C = \( 72^\circ \)
- Angle D = \( 108^\circ \)